I believe that the best sort of drawstring is no string at all--drawstrings have the horrible potential to come loose and strangle the baby.
Illustration 1: If drawstrings absolutely must be used--as in a family heirloom christening cap for example--their danger can be lessened (but not eliminated!) by sewing the drawstring to the cap while leaving the extending ends to be used as ties, but keeping these ties as short as possible.
Illustration 2: If the cap will be too large for the baby's head unless the drawstring snugs the cap up, then sew the drawstring to the cap in the already snugged-up position.
Illustration 3: I believe that a safer modification for a drawstring cap is to thread the drawstring through the eyelets, sew it down, and then work the protruding ends into a frog and frog-closure.
At heart, a frog is nothing but a knot, and a frog-closure is simply the little loop which slips around this knot, acting as a loop-buttonhole would. I think that frogs are safer than buttons, as they cannot be pulled (or bitten!) loose by a teething baby. Frogs come undone from their closures easily, it is true, but this is actually an advantage: you WANT the frog to pop loose with very moderate pressure, for safety's sake.
Addendum 11-18-09: I forgot to say: the illustration shows a single knot. However, if that is coming out too small, consider making the protruding end a bit longer, then folding the end back on itself, and THEN tie the knot in the doubled-back end. Also, make the frog-closure loop smaller than you think: it'll stretch out through use.
--TK You have been reading TECHknitting on drawstrings and baby hats.