How do you say jaslyn in Spanish?

There is no real spanish equivalent to that name. The closest I can think of would be Jacin or Jacian. In most cases, spanish speakers would probably just say Jason as 'Jazon' (Ha-Sone)

Maybe Jacinto, it sounds pretty close to me.

There is a Spanish equivalent for the English Jason: Jasón. Same spelling, with an accent (and stress) on the ó. See the Spanish Bible (NVI), Romanos 16:21. And yes, it is pronounced Hah-SONE.