Should I always train for basketball as hard and as fast as I can?

No. Continuous high intensity training takes a toll on your body and could lead to injury.  Although power exercises (like box jumps, medicine ball drills, and kettle bells) and max strength exercises may be important to athletic performance, it also important to work on balance, coordination, and stability.  Spending 4-6 weeks doing high intensity training followed by 4-6 weeks of stability and endurance training gives your body a chance to recover while working on another essential part of your performance.

You most definitely should not train as hard and fast as you can at all times.  Basketball is just as much about being able to slow down and stabilize, as it is about creating force, changing directions, and jumping.  Often times, injuries occur when players are unable to properly decelerate and control their movements.  For this reason, you must train through the entire muscle contraction spectrum of movement. This means that when designing a training program, it is a great idea to spend 2 to 4 weeks correcting any movement dysfunctions that your NASM Certified Personal Trainer has identified.  This is followed by about 4 weeks of stability training, then 4-6 weeks of strength and endurance training.  After these phases have been properly completed, you can add power, agility, and speed exercises that allow you to train hard and fast.  It is important to note, however, that you will still train to challenge stability, endurance, and strength during the days you are not working on power, agility, and speed.