Question: How can I pay my babysitter for working on Shabbos?
Answer: The Mishna Berura (306:16) writes that chazal forbade one from getting paid for services performed on Shabbos as they were worried that people would inadvertently do business on Shabbos. As paying one will cause them to transgress, one is forbidden to pay another, too (ibid 306:21).
R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 28:68) writes that one who received money would never be allowed to benefit from it. Quoting R’ Chaim Soloveitchik, he writes (28:n110) that if one worked on Shabbos intending to get paid, that work is forbidden, too.
Ideally, one should ask their babysitter to do some work for them during the week, and pay them for that work and their Shabbos babysitting as a single unit, rather than per hour (Shulchan Aruch OC 306:4). R’ Shlomo Zalman Auerbach held that this extra work must be something for which one would normally pay another to do (Shemiras Shabbos Kehilchasa 28:66, n145).
Likewise, the Aruch Hashulchan (OC 306:12) writes that one may pay another (an inflated fee) for any preparations they did during the week, such as travelling.
Another option is to give a gift, rather than paying them (Mishna Berura 306:15). R’ Ribiat (The 39 Melochos, Kosaiv p976) stresses that this gift must not have been pre-negotiated, and both parties must understand that it is indeed a gift.
In conclusion, one cannot pay babysitters normally if they just work on Shabbos. One may give them a gift for their work, though it is ideal if they do some work during the week, too, and are paid, even an inflated fee, for that.