There's little left of the old traditional covered suqs any more. The closest you can come is a relatively small mini-suq that is located along the Creek not far from the Dubai Museum on the Bur Dubai side of the waterway. Here, depending on the time of day, you'll find the usual crowded stalls with goods piled helter-skelter around. Although you may not find much of any real interest -- most of the goods are pretty cheap and meant for locals generally -- you will find plenty of atmosphere. And it certainly is a respite from the usual glitz and glamour of Dubai's amazing malls with their inflated, international prices. In terms of goods that ARE available here, it's mostly fabrics, incense, and (cheap) shoes, as I recall.
Why fabrics? The reason is most people do not buy clothing off the rack, but buy fabrics and take them, together with favorite garments, to a local tailor to have new items stitched up. The little hole-in-the-wall tailors found in every Gulf city are remarkably good. Indeed, many of my shirts have been made in just that way. If you have some time -- three or four days -- you could easily do the same. Just ask someone in your hotel for the name of a good local tailor. They'll know! As for price? About $7-9 for a shirt... Not much more for a woman's dress.