Brass or Stainless Steel Base?

Both materials are equally excellent, as you are holding the fixture, you can notice the brass fixtures are heavier than stainless steel fixtures. 

Brass - Brass is typically widely used for its great qualities, as much higher-end quality plumbing fixtures are produced by this material (also reflects within the pricing)

Brass is an alloy composed primarily of copper and zinc that may also contain small amounts of other metals. These additional constituents give the brass specific qualities for particular applications and vary by use of the end product. Overall, as a material brass is highly corrosion-resistant, extremely durable, and has greater workability than materials like stainless steel. This is to say, it lends itself well to forming into more detailed shapes for products like faucets and console vanity legs. Additionally, brass is a recyclable material and even has scrap value at the end of the service life of the finished product.

Stainless Steel - Stainless steel sinks present durable longevity as they are able to handle different kind of impacts and cleaning chemicals (just as resistant to rusts and corrosions as brass)

Stainless steel is a ferrous (iron-based) alloy with at least 10.5% chromium to provide oxidation resistance. Stainless steel is rust and corrosion resistant, highly durable, and offers excellent structural strength. Stainless steel offers these qualities at a lower weight than a comparable item manufactured from a material like brass. Finally, stainless steel can be recycled at the end of the product’s lifecycle. Unfortunately, the scrap or recycle value for stainless steel is quite low, but at the very least it does not have to go into a landfill.