Could it be the alternator?

The alternator not charging will cause the battery light to come on. This could be from the alternator simply failing unspectacularly, which would not cause smoke, or some other things. The bearings in the alternator may have seized, causing the belt to rub on the pulley and smoke. This will usually be accompanied by a noise, but not always. The belt may have snapped or simply come off, you will also lose power steering, which would be very obvious. The water pump is also driven by the belt, and the engine would overheat pushing coolant out of the overflow bottle causing a steam eruption. A couple miles is probably not enough time for this to happen, and there would be other warnings present with this, not just a battery light. A coolant leak on the alternator is not very likely to be the problem, the alternator is above the water pump on that model (I think), and it gets completely drenched whenever the van is driven in the rain. The most likely scenario is an internal short in the alternator causing the battery to discharge into the alternator windings and burning the insulation, This will quickly convert the varnish on the copper windings into smoke, and discharge the battery without overheating the engine or compromising power steering.

SOUNDS like the ALTERNATOR bearings are SHOT and so is the DRIVE BELT! >>You may ALSO need a NEW BATTERY As it is a FACT that they ALL WEAR OUT TOGETHER! the BATTERY, the ALTERNATOR and the VOLTAGE REGULATOR! <<<DO NOT get an AUTO ZONE BATTER>>> they are JUNK! go to the JUNK YARD and get a good USED BATTERY from a recent WRECK! A FACTORY BATTERY or an INTERSTATE or a DEKA is VERY GOOD! >>>>POWER WASH the engine with GUNK ORANGE original engine cleaner FIRST, if you CAN!>>> if not AFTER you fix the PROBLEM! OF COURSE open the HOOD and see WHERE the SMOKE is COMING FROM! PEP BOYS used to carry TERRIFIC BOSCH rebuilds that are BETTER than the FACTORY ONES and there are Alternators, STARTER repair shops in most CITIES as well! BOSCH is the BEST rebuilder HANDS DOWN! >>START by CLEANING the ENGINE and looking at the ALT belt to SEE if it is RED HOT or just FRAGMENTS hanging by a THREAD!  It MAY even be FLUID leaking onto the ALTERNATOR or BELT and That is the REAL PROBLEM! OR MAYBE even a leaking OIL from the ENGINE FRONT SEAL>>>BEGIN using PENNZOIL ULTRA PLATINUM OIL ONLY and get a WARRANTY out to 500,000 MIles and FORGET el cheapo OIL CHANGE ctrs! MOST use BARREL CHEAP OIL and PAPER oil filter elements! There is FLEECE and SYNTHETIC FIBER oil filters from PENNZOIL out there TOO!PENNZOI Lis made from NATURAL GAS and RPEVENTS sensor failures and SEAL and GASKET LEAKS and RUBBER =breather hoses from MELTING and Decomposing! >>>ALWAYS keep the ENIGINE As CLEAN as possible so HEAT will DISSIPATE as it NEEDS TOO! ENGINES are REALLY like PEOPLE In a way! The METAL needs to BREATHE to be HAPPY ! The GOAL is to have it CLEAN on the OUTSIDE ( gunk engine cleaner) and CLEAN on the INSIDE (PENNZOIL ULTRA PLATINUM) or ULTRA EURO that is good to 35 BELOW ZERO! and gives a 500,000 mile WARRANTY! It keeps SENSORS and the ENGINE INSIDE as CLEAN as the DAY it was MADE in a few short weeks!  GOOD LCK but you will NOT really need it! 

A bad coolant leak that got onto the belt and the alternator pulley was slipping and not charging, so the battery light came on. This is just one possible scenario that fits your evidence. 

Another possible scenario is that your belt came off.

Thinking you cooked the battery.  See a mechanic.

Sounds like you def have some car probls

Open the hood.  Serpentine belt could be broken.  Good luck.

Did you open the hood and look?

Not familiar with your car's set-up:  does one belt turn both the alternator and water pump??

The alternator would make the light come on, but not the smoke (steam?)  I presume that if the engine was hot, you would have mentioned that .  Right?

What would the white smoke have to do with an alternator? Nothing. Sounds like maybe a hose is leaking antifreeze, that would be my first guess. Why not open the hood and take a look to see what may be going on.