Instructions
Practice playing scales on the bass. This will utilize all four fingers on your fretting hand so that you can learn to transition more quickly between fingers.
Play for about 30 minutes each day to develop the calluses on the fingertips of your fretting hand. Calluses improve the strength of your fingertips and create a crisper sound with less effort. This helps maintain energy in your fingers for fast picking.
Utilize all fingers and thumb except the pinkie on your picking hand. Play the first note with your thumb and then follow that up with your fingers. Beginner bass players often use only one or two fingers when picking, but incorporating all digits will increase your speed.
Rotate among strings as you pick with each finger. Work with the same string pattern to help develop accuracy when playing -- you will become familiar with finger, hand and forearm positioning for each string.
Don't attempt to play faster than you can. The key to developing speed is playing each note clearly and slowly, if necessary, and then increasing your speed.