IS THE VEHICLE IN PARK? IS YOUR FOOT ON THE BRAKE? Then it might be the neutral safety switch located on the side of the transmission. It's a fairly inexpensive part/repair.
Did you put a starter on it ? If so, Then you have put the starter wires in the wrong place are you have the big battery wire touching the small wire that goes to the starter. Recheck the wiring on the starter.
starter motor need overhaul
The starter, on your 1998 Dodge neon, is located near the bottom, back, left side of the engine. You can follow the battery cable from the battery to the starter.
Had the same problem. Bought a re-conditioned alternator, turned out it was bad. Usually if battery draining, it's alternator or battery or both.
The starter solenoid is attached to the starter. Follow the positive battery cable from the battery, it will lead you to the starter solenoid. The starter would have to be removed to replace the starter solenoid.
dead battery or starter
Remove the positive cable from your 1998 Monte Carlo battery. Remove the cables from the front of the starter. Remove the starter retaining bolts. Reverse the process to install the new starter.
check the battery or battery cables or the alternator or starter
Starter Relay? Yes, under the hood in the black box by the battery
Follow the + battery cable its the small part of the starter where the wires hook up
follow the positive wire of the battery and it will lead to the starter becareful not to touch it because it will shock you
Remove the positive cable from your 1998 Chevy Cavalier battery. Remove the cables from the front of the starter. Remove the starter retaining bolts. Reverse the process to install your new starter.