Drawstrings are seen as dangerous because of the potential for them to catch on busses and play equipment (yes, it really happens!).
Before we jump in, what is the definition of a drawstring?
"The standard defines a drawstring as "a non-retractable cord, ribbon, or tape of any material to pull together parts of upper outerwear to provide for closure."" (CPSC, Drawstring FAQ)
Federally in the US, the Customer Product Safety Commssion (CPSC) have banned drawstrings on upper outerwear for sizes up to 12 (children's). Since this is only upper outerwear, they are looking at coats, vests, hoodies, cardigans, etc. that would go over another top and be worn outside. I would also be cautious of hats with strings, but the CPSC allows it stating they should only be as long as necessary to function, but prefer elastic, snaps or hook and loop material be used.
The CPSC also excludes garments like dresses, shirts, pants, etc. with the same guidance that the cords and strings are only as long as necessary to function. Belts are not considered in this regulation either.
The CPSC also wants to see no knots in the drawstring since knots are easily caught. In retail stores, you'll see either an aglet end (plastic or metal), or that weird folded and barely tacked-down method. These are less likely to be caught and if they are caught, more often will break.
Alternatives to drawstrings can be hook & loop material, snaps, buttons, and fully-attached elastic.
A couple of states (WI and NY), as well as Canada and Europe require drawstrings to be tacked at the back center and on the ends and be no longer than 3" from the opening of the channel.
Now, faux drawstrings, there aren't any regulation on these because if they were to get caught, they would break off of the main garment instead of keeping the wearer at the caught point. Even still, we don't want to leave room to catch on things because we don't want to harm the garment! So, I do recommend following the same process of making sure there aren't any knots and instead of a smooth ending to the drawstring. I also recommend keeping it closer to 3" in length, but again, since faux drawstring do not pose a safety risk, this is a personal recommendation.