There are currently two main types of semikhah. One is called a heter hora'ah and the other has no real name but is colloquially referred to as a rabbi's driver's license. The former is a license to issue halakhic rulings. The latter is essentially a letter stating that the holder is worthy of holding the title rabbi and being a leader (manhig, which can also mean driver). I do not know the historical origin of this type of ordination but I know that it was given in Hildesheimer's Seminary in Berlin (with a heter hora'ah only given to the most accomplished graduates) and, to my knowledge, is what JTS has always given. YU gives a heter hora'ah. Hayim Berlin and, in the past, Ner Israel give the driver's license. The question we will address is whether either of these two types of ordination may or should be given to women.
This is not a new question. It was addressed extensively in the 1980s by scholars of the Conservative movement, and my presentation here is generally informed from those analyses.
To my surprise, I did not see any Conservative scholar raise the issue of serarah. As we discussed in regard to converts, neither they nor women may be appointed to positions of communal authority. The majority of contemporary posekim rule that women may not be appointed to such positions. Therefore, it would seem to follow that women may not serve as pulpit rabbis which, presumably, is the quintessential communal position. R. Moshe Shternbuch addresses the question of whether a convert may serve as a pulpit rabbi and ruled that he may not but, R. Shternbuch suggests, someone else should be appointed to that position and the convert can have a different title while fulfilling the role but not the position of the rabbi. While this is informing, it is not a viable model to be used on a large scale.
II. Rabbinic Functions
A woman may not perform many of the functions normally performed by rabbis in many synagogues. For example, rabbis, particularly in small synagogues outside of the main metropolitan areas, are called upon to lead services and read from the Torah. Almost all rabbis are called upon to perform weddings. Women may not perform these functions. Thus, a woman is automatically prevented from taking a large portion of rabbinic positions.
If a woman cannot be hired as a pulpit rabbi and cannot fulfill some of the functions of a pulpit rabbi, it seems clear that a woman cannot be ordained to be such a rabbi. Therefore, the driver's license permitting one to serve as a communal leader should not be given to women. There is another option, we will get to at the end.
III. Rabbis and Judges
The heter hora'ah is part of a chain of a centuries-long tradition. As Prof. Saul Lieberman demonstrates in his responsum on the matter, semikhah was traditionally given as permission to rule and to judge. Even today, when there are certain institutions that give special ordination to serve as a judge (yadin yadin), to my knowledge it is standard practice (at least in the US) for rabbis lacking such special ordination to sit on rabbinic courts. The heter hora'ah is considered to be sufficient for serving as a judge.
Women, due to their exemption from forced testimony, are barred from serving as judges. They may not sit on any official rabbinic court, even though they may judge on an ad hoc basis when both litigants accept them. Because of a woman's inability to receive ordination for judging, she was barred from the official type of ordination that dates back to Moshe. Additionally, a woman should not be ordained because she cannot sit on an official rabbinic court and the heter hora'ah is considered license to do so. Even if we choose to disregard the historical requirements of "true" semikhah, which is probably not the ideal approach, we may still not give women the standard semikhah that is used today because it is considered license to sit on a rabbinic court.
IV. Practical Considerations
It could be argued that women who are worthy of the title "rabbi" should be trusted not to act improperly. They are certainly religious and will therefore only accept rabbinic positions that will not force them to compromise their values. Even if so, granting women rabbinic ordination in which they will be severely limited in their professional options will create an extremely imbalanced field of employment that will be rightly called discriminatory and will put unfair economic pressures on female rabbis. If the current situation is considered unfair, the resulting situation will appear to be much more discriminatory. Men, some significantly less capable than their female peers, will have greater employment options and more opportunities for professional experience (thus leading to pay imbalances). This situation will not last long and women will inevitably begin taking rabbinic positions that will cause them to compromise (or, more likely, change) their religious values. The ordination of women based on very specific limitations will unquestionably lead to the erosion of those limitations.
V. Communal Considerations
It must be remembered that a substantial segment of the Orthodox community considers the advanced learning of the Oral Torah by women to be forbidden. This is not an obscurantist position but a well-established halakhic view that is amply supported by traditional sources. This segment of the community cannot accept female rabbis or Torah teachers. The ordination of women will effectively cut off all ties to this portion of the Orthodox world. Communities with female rabbis will be considered beyond the pale of Orthodoxy. Setting aside the not-altogether-meaningless subject of denominational titles, is this further fragmentation of Kelal Yisrael a positive outcome? Should we not be striving for communal unity rather than driving unbreakable wedges between our sub-communities?
VI. Confirming the Heterodox
In an earlier series of posts, I outlined the prohibition to confirm the Heterodox in their positions. It is very likely that ordaining women as rabbis falls under this prohibition and, therefore, must not be done. The contra-halakhic trends of egalitarianism are still very much with us and we may not support them, even indirectly or unintentionally. Let us not be naive about these very real matters.
VII. Other Options
Rather than ordaining women as rabbis, the most viable suggestion is to create a new title that reflects advanced scholarship but lacks the history of the title rabbi. This title will designate women as scholars who are qualified to teach. There might even be differing levels, including a teacher and an halakhic scholar. I see no reason why a woman cannot rule on halakhic matters to those who ask her, even if others will ignore her rulings. If institutions start offering programs to grant women the titles of melamedet (instructor) and poseket (halakhic decisor), I suspect that they will be somewhat successful. I doubt that they will be graduating posekim on the level of R. Moshe Feinstein, who, in addition to his inherent talents, was preparing incessantly for his position since his early childhood, but they will certainly train women qualified to rule on basic matters. Certifying an advanced melamedet is certainly uncontroversial. Regarding a poseket, though, the question remains whether this is a confirmation to the non-Orthodox and, therefore, should be delayed to a generation that does not have the same recent past as ours.
A further consideration that I initially neglected to raise, but which I consider to be of primary concern, is that of tzeni'us. Should women be rising to roles of public leadership, not to mention speaking publicly in front of large gatherings of men? This is one that is mixed with both halakhic and meta-halakhic issues. Because I assume that those on "the left" will automatically reject such considerations and those on "the right" will consider them decisive, I will leave the discussion of this matter for a later date.
 See David Ellenson, After Emancipation, p. 189 n. 11; R. Yehiel Ya'akov Weinberg, Seridei Esh (Jerusalem: Mossad Ha-Rav Kook, 1969), vol. 4 p. 133.
 For an extensive review of the literature, see R. J. David Bleich's chapter on women serving on synagogue board in Contemporary Halakhic Problems, vol. 2
 Teshuvos ve-Hanhagos 3:305
 That a woman may not lead services and read from the Torah was a major point of debate in the Conservative discussion but the only serious voice among the lenient, Prof. Joel Roth in his "On the Ordination of Women as Rabbis" in On the Ordination of Women, did not, in my opinion, make a convincing case. Prof. David Weiss Halivni, in his unpublished article on the subject available in the JTS library, argued that women may not perform weddings. He was not particularly convincing but see Rema, Yoreh De'ah 242:14 (and Pis'hei Teshuvah, Even Ha-Ezer 49:2) and set aside the circularity issue. If women cannot be rabbis for other reasons then they may also not perform weddings.
 Printed in Tomekh Ka-Halakhah, vol. 1
 See the discussion in this post regarding converts on this issue
 Prof. Lieberman writes, "[W]e should not empty the title 'rav' of its meaning from the way it has been understood by the Jewish people throughout the generations." But note that his institution only gave the "rabbinic driver's license" and not a "heter hora'ah"!
 "Adoption of Heterodox Practices" I, II, III, IV
 See the responsum of R. Moshe Shternbuch in note 3 regarding a convert ruling on halakhic matters.