Concur with cold water wash and white vinegar. One other thing that you need to do is to look into having a french drain and a sump pump installed in your basement. (I'm assuming that the basement floor and walls aren't finished - but either way you probably need the french drain and sump pump.) That's somewhat costly, but will also prevent mold and mildew, which is far more costly in the long run.
We have an older house where we had to redo the entire basement floor and add a french drain/sump pump. We got estimates between $23,000 with a "2good2btrue clause" that guaranteed we'd get all our money back in 3 years (probably took sales training from Bernie Madoff!) and $8500. We went with the contractor who estimated $8500. He did a terrific job. The contractor advised us to drill weeping holes in some of the cinder blocks of the wall, and to regrade the land on one side of the house. The weeping holes in the wall allow water that comes in from outside to quickly drain to the french drain instead of slowly seeping through the cinder blocks, which works as a food factory for mold! Haven't had water problems since.
Just to make you laugh a bit...while the contractor was prepping the floor, he took the pole that the washing machine drained into out of the floor. He came out of the basement with a puzzled, slightly amused look on his face. He handed the pole to my husband and said "There is no drainage pipe in the floor for this, and it's not hooked up to your plumbing. Your washing machine has been draining into the basement floor for years!"
No wonder the floor was such a mess!