How can I protect my fingers while using a band saw?
Do not wear gloves while using a bandsaw (or any powered saw, drill press, or planer for that matter). The gloves will give you a false sense of security and do very little to protect your hands. In fact, they may end up getting your fingers pulled into the blade if the material catches. Which would you prefer: a cut fingertip, or a mangled hand?

The best way to protect your hands is to be very deliberate about every movement you make around the saw, and keep your hands away from the blade by using a push stick or push block. If the cuts you are making are simple straight lines, get a miter gauge that slides along the miter slot that is present in most band saw tables.

Assorted Push Sticks:



Miter Gauge:



EDIT:

After a clarification from @ArgentoSapiens, I realized you're looking to cut the smaller person-shaped game pieces and not the board tiles. The way those are most likely cut in the factory would be with a router table and specialized router bits. Stock would be fed through to create one very long "extruded" meeple which could then be cross cut into many smaller pieces that are only 1" or so thick. You could probably do this with just a few router bits if you've already got a router and a table. This would be a much faster and safer way of doing it.

If you don't have a router and router table, then I suggest using a band saw or scroll saw as you planned. However, don't cut the stock into roughly "meeple" sized pieces first. Plan on wasting a lot of wood for safety's sake ($3 for another 2x4 is well worth keeping all of your fingers intact). Keep each work piece attached to a larger chunk of wood that you can use to safely feed the rest of it into the band saw. Then when you have all the detail work completed, finish up by making one last cut to sever the finished piece from the larger chunk of wood.