I detect confusion here, and I'd recommend you take some time on the bitcoin wiki to clarify:
> "If the rewards go down over time, eventually it won't be profitable to mine bitcoin."
> Yes.
No. Because the other part of mining is that you get transaction fees for it. Mining isn't a luxury, it's necessary for the network to function (otherwise double-spending isn't prevented). So as time goes on and less coins can be made by mining, transaction fees will increase to compensate and keep mining profitable.
> Also, computational power gets cheaper with the passage of time.
But, mining does not get cheaper because the network (by design) adjusts the mining difficulty to keep the incoming block rate constant. So in relative terms, mining will never become significantly more or less expensive with time.
> The largest caveat I see there is the part about more of them being demanded.
Not a problem. This is solved because bitcoins are infinitely divisible (up to 8 decimal places currently and this can be extended if needed). So if there's more demand, the value of coins increases but you can still trade them in whatever meaningful quantities you like. This isn't something you can do with cents, but you can trade one millionth of a BTC just fine.