So, my next step for me would be to compare the input and output on the scope and see if they match or assume it's fine since the output isn't floating?
Even with an oscilloscope, you can generally only check one leg at a time (some scopes have dual probes, but it's harder to keep two probes in place than one). But, the oscilloscope will show you a visual pattern, and if you see the same pattern on your input and output, then you can say things are working (so much better than audible beeps!). Again, the LS257 output pin depends on the state of several input pins. If, in the game, the select line is transitioning between High and Low, and you have two probes (1 on input A, and one on output Y), then they won't look the same, because at some point input pin B was selected, and that changes the output to match B during those intervals.
Generally (very generally...), if your outputs are changing, then the chip is working. If you see an output not changing, that is a potential for a problem. But there are lots of outputs that stay High or Low because that's how the game designer designed them, or becase a particular output pin is not used, or is tied to GND, etc. You're on the right track, if the output is correct for the given set of inputs, then the chip is probably OK. If the output doesn't match the truth table of what is supposed to be going on, then the chip is suspect.
It's really necessary to understand the schematics of the game, the logic of the chips, and how they relate to make the final determination.
So, rather than looking at the LS257, look at the chips supplying the inputs. My gut at this point (without knowing anything other than what I've heard here...) is that your select line should be toggling, at least at some point (assuming your game is running - if your game is hung, then I would expect to see nothing changing on the outputs). Maybe it is, and your probe just isn't seeing it, or maybe it's not. That's the whole point of the LS257 -- to select between lines A and B. If you are only seeing one or the other, then figure out which output pin on what chip is feeding that select line input pin, and test that chip's output to see if it is stuck high. And, if it is, look at the schematics to see if that is a design "feature", or a correct result based on the inputs for that chip. Work your way back through the chips until you find one that isn't working properly (assuming that a chip is the problem..)
Sometimes, its easier to understand what's going on with an Oscilloscope. And, at other times, it's just too damn time consuming - like when testing a flip flop. It's nearly (or is...) impossible to test a working flip-flop with an oscilloscope one leg at a time. That's were a logic comparitor (like the HP 10529A) comes in - simply install a duplicate, working chip into the comparitor, and clip onto the chip on your board, and bang - instant feedback on whether the chip is good or potentially bad (again, some chips appear bad when they aren't due to the game designers grounding output pins, or using chips in some other creative ways...). The down side to the HP10529 is that it is only capable of testing Dual In-line chips with 8 pins or less per side. If you need to test a larger chip (longer, or wider), you need something else. Also, some newer boards (like boards made in the 90's) have their chips so close together that you can't get the HP's logic clip on the chip.
There's no real replacement for an Oscilloscope though - extremely useful and versatile.
If you want to describe what game you are working on, and what symptoms you are seeing, that might help us to give better suggestions...