Welding is fine -- in essence it just creates a cheap, do-it-yourself spool that locks both sides of the differential together permanently.
Don't try the "fozzie locker" setup advocated by some (leaving some travel in the spider gears before they hit the weld spots). That is just plain stupid. If the spider gears can't spin all the way, they will just start spinning, gain momentum, then snap something. Better to be solidly anchored from the start.
The drivability issues have already been noted. What happens here is the inside tire in a corner needs to spin at a different rate than the outside tire due to the fact that the circumference of the arc of travel for each tire is a different length. Normally, something in the differential unlocks and allows each tire to rotate at a different speed to make up for this different amount of travel, allowing one tire to spin faster if need be. When both sides are totally locked together, one tire MUST drag going around EVERY corner, no matter how fast or slow you take it. If you hit the throttle at the same time one tire is dragging, you will probably break traction on both, and with both spinning, you WILL slip sideways. This is magnified on wet or slippery roads. So, while you have double the traction going forward, you also have half (or less) the traction in corners once the other tire breaks loose. Once welded, there is no remedy for this except to change out the entire carrier.
Otherwise, welding works well off-road. It is cheap, doable at home, and effective.
As far as the actual procedure... Start by CLEANING all the metal parts as good as you can. I usually use brake clean spray. MAKE SURE that you give the brake clean adequate time to dry before laying on the weld, or you will have a nasty little fireball pointed right at your face.
Second step is to pre-heat the parts with a torch. They should be warmed up to just under dull-red. Then, using a good grade of rod (I like to use US FORGE "Steel" or "Problem Solver" rods with stick welders) start to weld each side gear to its mate, and if you want, the side gears to the carrier also (strongest, but ruins the carrier). Make sure that you can still pull the center pin so that you can get the axle shafts out, and also, as mentioned above, that you don't weld the C-clips in. It doesn't take THAT much weld to make a successful job. Just locking the side gears together will suffice to make it into a "Lincoln locker."
