Anti-Lock Malfunction?

I have a 2001 Silverado LT and occasionally when I apply the brake at a low rate of speed, the brakes will not work. I have to tap the brakes and then the brake will work. It doesn't appear to happen with sudden stops or driving over 20 mph. Someone suggested the anti-lock sensor needs to be cleaned or replaced. I've been told that the repair could be upwards of $1,200.00. Has anyone experienced anything similar & what was the outcome?

Thanks!!

It cost approxamatly 2-300.00 at the dealer. All labor and it has fixed mine now for 15000 miles. This will occur again as i live in the snow belt. Next time i will do it myself. These wheel speed sensors are expensive and somwhat fragile so its a toss up, as to the cheapest corse. Your post said over 20 mph ill assume you meant under 20 mph or you may have a different prob..JJ