Rav Moshe Feinstein was asked in 1961 about a woman whose doctor had artificially inseminated her insofar as she and her husband had been unsuccessful in having children and the suspicion was that it was a deficiency on his part, not hers. The question raised was whether the woman would be forbidden to her husband like an אשת איש שזינתה and what the status of the child would be.
In terms of the woman, Rav Moshe ruled that she would still be permitted to her husband, since a woman is only forbidden to her husband as the result of having illicit sexual relations, which did not occur in this case. The mere presence of another man's semen in her uterus is not, in Rav Moshe's view, sufficient cause to force her to separate from her husband.
In terms of the child, Rav Moshe is similarly lenient. He postulates that the child would only be a mamzer if there had been forbidden relations, but since there were not the child is thus כשר. As proof, he cites the case of Ben-Sira, who was allegedly conceived when his mother absorbed a man's semen by entering a bath after he had been in there - and yet Ben-Sira was considered to be ולד כשר. Even further, Rav Moshe rules that even if the doctor claimed that the semen was from a Jewish man he is not to be believed (this is assuming a non-Jewish doctor in America), since we assume that he only said that knowing that the woman was Jewish and that she would have preferred that the donor be Jewish. However, since most donors in America are not Jewish, we can rely on that רוב and the child would be permitted to marry anyone. If the child were to wind up being a girl, Rav Moshe is prepared to allow her to marry a kohen, demonstrating just how far he is willing to go with this line of thinking.
The only way in which Rav Moshe sees some deficiency in the lineage of the child is that he absolves the husband - who is not the father - from having to provide for this child or from having to pay for the delivery, although that latter ruling is due to the fact that he was not complicit in the decision to inseminate the woman.
(Note that unlike the teshuva posted from Rav Sherlo on this topic, in this case the woman was married and had not succeeded in conceiving. I am not sure what Rav Moshe would rule about a single woman who wanted a child without marrying - but stay tuned.)