I voted for Fluttershy, she seems to be perfect for a baby sitter as she takes cares of many different kind of animals.
It didn't go too well where she had to take care of the Cutie Mark Crusaders, but it worked out perfect in the end.
Apple Jack - Eh, she seems pretty busy picking up apples from the trees.
Rainbow Dash - Too cool to handle baby-sitting.
Pinkie Pie - Another pony that's pretty good with baby-sitting, didn't go too well but worked great at the end.
Twilight Sparkle - Mature and responsible, I think she can handle the baby-sitting as-well.
Rarity - Too busy with clothing, wouldn't care/give a single f*ck.