The only way I would think you're at risk is if you use anti-static dryer sheets. I could see there being a problem with particulate matter coming off those (it has to, after all, to be effective on your clothes). I wouldn't want my dd breathing it in. But as long as you don't use them, I wouldn't think it's much of a safety issue. What I would worry about long-term with is is moisture build-up in your basement, because you're effectively transferring all the moisture from your damp clothes into the air in your home. You will likely see a mold problem start around where the vented moist air goes. Mold CAN be a big health hazard, but it probably needs a couple weeks to get going.
Postpartum doula & certified breastfeeding educator, mama to an amazing girl (11/05) and a wee little boy (3/13).