Do sperm banks allow the woman to preselect the characteristics of the donor whose sperm she would receive?

My question was prompted by a fanciful thought of getting pregnant by an unknown man during blindfolded sex. Of course I don't go for that lifestyle, so the nearest fantasy is to be made pregnant by an unknown sperm donor, but one whom I would be told something about on the lines you've mentioned.

I have a fanciful notion that you live in India.
If that's the case, what would the reactions of your husband, relatives and neighbours be?

One way that lesbians sometimes choose to get pregnant is to choose a trusted male friend,
ask him to donate his sperm.
With everyone nearby so there are no time lags, he does it into a condom, gives it to one of the women and she uses a syringe (without a needle) to fire it up into her beloved.

Yes, I do live in India, am childless and never married. I have very understanding parents who have stood by me through my relationship upheavals, though let me clarify I'm by no means promiscuous. I live in Gujarat in a very conservative (upper) middle class society. My very few relations have been very supportive too, but I can't say the same for most of my parents' circle. I've lived on my own in a separate flat since I was 24, though I'm close enough to them should they have an emergency. I'm very close especially to my mother with whom I share my most intimate secrets. I know I can do the same with my father too if need be, but somehow haven't gotten myself to do it except in a couple of instances. Both my parents are mygems, and they've always considered me as their gem  My name, Neelam, means sapphire in Gujarati. I dread the day they won't be around. My father is 74 and my mother 69, closer to 70.