Yes! Use whatever medical options that are prudent to ensure that those who are on welfare for more that a very short period are prevented from conceiving children at least during the time they are on welfare.

Are you talking about mothers, or fathers as well (bearing in mind that a few stereotype no-hoper males can create another generation of schoolgirl pregnancies who can then go onto welfare in the third trimester or with child in tow)

You describe "prudent" options. well, the only means of ensuring long-term male contraception is surgical- requiring an operation for every male who goes on welfare and another when they go off it, with compensation for any male who experiences the known complications of surgery imposed against their will?

As for long-term female contraception that cannot be forgotten or voluntarily withdrawn: the IUD is unsuitable for the sort of lifestyle we would assign to the stereotype no-hoper female and can fall out or be removed fairly easily, so that leaves surgical options (see above) or long-term hormone options like depo-provera. However, it can take 18 months after depo is withdrawn before a woman becomes fertile again, leading to the potential for juicy compensation claims again.

Not trying to be facetious, just observing that there are no easy answers and that the sort of person who is determined to access welfare has plenty of time to invent loopholes. But as long as they are the minority, IMHO we need to grit our teeth and carry their miserable load so we can be sure that those who actually need welfare are subject to as few demeaning policies as possible.