Yes, it is due to feminism. Men have no say about anything concerning legal/financial matters of their sperm. Have always believed there would be fewer unwanted/unprepared pregnancies if single men were legally able to 'choose' not to be a parent like women do. Equality is not.
In the case of sperm donorship, I think the father chose not to be a parent, just helping someone out who wanted a kid.
In other cases, it's complicated. Can a man tell a woman to have an abortion, because he doesn't want to be a parent? Can a man tell a woman, who, unplanned, fell pregnant tell her to get the baby, because he wants to be a parent?
Ideally, there would be communication on this. But if the final say would be with the man, rape could become the way to get offspring.
I don't know. Feminism comes in many forms, as does socialism, capitalism and what not. If it can be used for benefit, every ideology will be bent to get that.
Feminist influence in my country means shared custody, by default, after a divorce. It means the richest of the two partners, after a divorce, will have to take financial care of the ex partner, no matter what gender. And, alimony for a partner will never be longer than 12 years. If the marriage hasn't lasted 5 years, alimony for a partner will not exceed the number of years the couple was married.
The maximum of 12 years will probably be brought back to 5 years.
Alimony for an ex partner makes sense. I know couples where there was the agreement they would study in succession, while the other provided the income. If after 6 years one partner got a degree, and a good job, and the time had come for the former provider to quit a job and get a degree, and divorce happened, the one with the degree better pay for the then unemployed and studying ex-partner. The degree had been paid for by the ex-partner.
Trying to pluck money from a sperm donor, well, a state who wants that done can use all kinds of ideologies to get that. Interest of the child, to know it's biological father, and the right of the child to know it's father cares, at least in a financial way, is a nice one to use. 
The "it isn't how god/nature intended children to be begotten" is another one.
Marriage makes things completely different and more complicated; just really talking about people who aren't legally bound to each other. Once read a case where a man and woman had a prenuptial agreement not to have children, but when she became pregnant without consulting him, the agreement was superseded by governmental laws. Only talking about single people, and not talking about men telling women what to do with their bodies. The donor's intentions don't matter against the definition of the law.
Sex will become very complicated. Condoms will be the easiest part. There have to be negotiations and legal documents signed. What to do, in cases of unwanted, unplanned pregnancies. Will there be an abortion or not? Will there be a baby or not? Will it be given up for adoption or not? Who will pay?
Don't think it will bring down teenage pregnancies. Nor that it will improve the fun in sex.
