Here are the steps to clean the steel strap
Step 1: Detach the bracelet from the watch
Different stainless steel watches have different methods to take bracelet apart from watch. Some watches have quick release bar that we can separate the watch head and watch strap with the click of a button. Some watches may need a special screwdriver to remove the bracelet from the watch head.
Step 2: Soak your watch bracelet
Immerse watch bracelet in a small bowl filled with soapy water or alcohol. Soaking your bracelet in this way will loosen the accumulated grit and dirt. The length of time to soak a bracelet depends on the dirtyness of the watch bracelet，ranging from 15 minutes to 1 hour. If you can't remove the watch bracelet from the watch, please keep the watch head away from water，especially when your watch is not waterproof enough. When the watch is completely waterproof, it would be fine to soak it for cleaning, but the only issue I think you might have is the waterproofing wearing down.
Step 3: Brush the link of the bracelet
Pick bracelet out from the liquid, use the toothbrush dipped in soapy water or alcohol to gently scrub away discoloration or dirt that has collected within the gaps of the watch band.
Step 4: Rinse with clean water
Rinse soapy water and alcohol away with clean water.
Step 5: Dry your watch with a soft, lint-free cloth
Dry your watch with a soft, lint-free cloth. Even after patting your watch band down with a dry cloth, it’s possible that some liquid remains between the links and crevices of the watch. To ensure your watch is totally dry, air dry it by laying it on a dry towel for an hour or use a hair dryer on a low-heat setting to quickly dry in between the links. This will prevent liquid from getting trapped in the watch band, limiting your watch’s risk of rust and corrosion.
Stainless Steel watches are mostly used for business, showing a sense of rigor and seriousness. However, if the steel strap is dirty and not cleaned up in time, it would be harmful for not only watch but also your impression. Did you clean you the stainless steel strap?