If your levels are all in one map, then the box triggers are probably a good route to go.
If each of your levels is a new map another possible method is to use multiple game modes (Level1GameMode and Level2GameMode) where the game mode for level 2 references an alternate character class than the one in Level 1. What you likely would want to do for your case is have the fully functional Level 2 character be your base class, then derive your Level 1 character from that base class and disable jumping and lower the max walking speed in the Level 1 derived character.