If the rails rusted away, then water was getting underneath the lip. I'm guessing the countertop rotted, and you replaced that too? That's why you mention cutting the hole?
Anything that allows ANY moisture through this gap will cause big issues, so you want something really good here. Definitely not a place for putty.
If the sink is thin (i.e. flexible), and/or not well bolted down, it can move enough to cause caulk to pull free, and then water will wick through, and your countertop rots. A really adhesive and flexible product here helps with that situation.
Silicone is ok, and works well for a thicker caulk line, if the sink or countertop isn't absolutely perfectly flat. But if the sink is pulling too much on the bond line, or if it is permanently wet, silicone may lose its grip.
Latex based caulks are most certainly not ok here.
Lexel is a good step up, if you want something clear (which should be fine unless you need to hide something), and can get a paper thin caulk line. This wipes clean with a handful of paper towels and mineral spirits.
If you go the marine store, they sell polysulfide caulk products that are much more friendly to work with than polyurethanes. That's another good option, and they come in a handful of boat themed colors.