There are a ton of houses for rent downtown on 5th and 7th avenue, between 57th and 75th streets. You could definately find something for under a grand per month. There are 5 on my block! However, before moving to Kenosha, keep in mind the following (especially in areas with a lot of renters):
-It's loud all summer. People blare their radios all hours of the night and stand out in their yards talking really loud.
-For some reason here, people here are in love with fireworks and set them off all summer (despite the fact they're illegal in town).
-There seems to be a large number of unemployed people milling around all day. But I don't know, it could be a lot of them work at night...
-Kenosha, Racine, and Milwaukee schools do pretty poorly in national standardized testing.
-Drug use and alcoholism are rampant, and seen as virtues.
-They don't plow the roads properly.
The beaches are decent (clean and large), especially Simmons Island; the water is freezing, but if you're from Sheboygan, you're probably used to it
I would look in Pleasant Prairie, Somers, or Bristol if I was you.
My wife and I moved here 2 years ago (relocated by company). This spring, they agreed to relocate us again when I told them that I was leaving Wisconsin, with or without the company. I travel extensively for work, and have been in pretty much every region of America for extended periods of time. Kenosha (city) blows. The taxes are insane, and the infrastructure the taxes are supposed to support is horrible. It seems like the town is trying to improve, and yet the improvements seem to be met with scorn by the life-long residents.
It's the only place I've ever lived or been in where I (literally) had to defend my family at gunpoint from someone else. Check the news; for a small city, there are a lot of assaults (phys and sexual), domestic disturbances, and the drunk-driving arrests could have their own section of the Sunday paper.
There will be plenty of people who disagree with what I'm saying. Take their point of view into consideration also, but with kids, I'd stay out of Keno and live out in the county.