When I was a grad student (in English), I tutored a nice family's two children twice per week in exchange for the babysitting of my toddler twice per week. I could have not afforded this invaluable time to connect with my spouse alone otherwise! Please do not underestimate the poverty and loneliness of your local grad students: underpaid and often far away from their extended families, such partnerships can be beneficial to both beyond mere money. I am a professor now but everyday I miss my "family" - seeing the children’s' improvement in school was a personal joy as was their seeing my little one grow. And beyond a few small Christmas gifts, all we ever exchanged was time, care and effort.

You might want to contact your local university or, as my family did, the English Dept or department of your choice directly. Hang a poster, ask around -- we may seem pretentious on the surface, but trust me, we are all human underneath :)