Historically basket hilts were introduced during the 16th century as plate gauntlet use declined. They were only used on swords. Certain types of guards (such as the cup hilt) did not appear until the mid-17th century.
Basket Hilts in the SCA
Other types of hand protection which can be used are ice hockey gloves (aka 'gumby'), gauntlets or clam shells. A quillion can be used instead of a basket hilt so long as any of the above hand protections mentioned above are used.
The pros for basket hilts is that they give your sword weight at the bottom, and are very effective in protecting your hand from injury. The disadvantages of using a basket hilt is that they can be confining, weigh your sword down, and some find it restricts wrist movement.