I need some help please.
My son Car won't start
Autozone test and have the Code P0335 crankshaft position sensor error
yes it start with jumper, well I will replace the sensor tomorrow.
Now the car start and run with the jumper if I remove the jumper then the car will stop, Put jumper back start the car and Check Voltage at the Alt. 13.89VDC but at the Battery only read 12.40VDC is that the cable may bad? or what els could be wrong? any help here thank you so much.

Forgot,
the Alternator not charging the battery.

You just answered your own question. Replace the alternator.

At the least take it off and have checked at Autozone or parts store with a checker.

I would double check the battery! If there is 13.8v coming from the alternator, then it should be good. Also check your battery cables for any corrosion especially the positive terminal. A shorted battery will cause all kinds of issues.

Get the ckp sensor replaced and get a dvom to check for a voltage drop between the battery and alternator. Also check for a voltage drop on body ground from the battery.

were you checking the voltage with the jumper cables connected if so you are reading your altenator output from your vehicle you used to jump start it.

ALTERNATOR AND RETEST!!