On the river, the situation usually becomes scary.
This is because everyone gets a downcard, so you wouldn't know if your opponents made their draw or not.
You might of had the best hand on sixth street, but since your opponents will be drawing to beat you, the tables could turn during the river.
Most of the time however, you usually need to bet anyway and see what happens.
The alternative to betting is when you are going low, with other players competing for your side.
If you think that there is a chance that they could outdraw you for a better low, you should just check and call.
This is because if you bet, the high hand might raise and then a better low might raise and you'd be pressured to call two bets on a hand that might not win.
If you have the best low hand however, and your are against other players with made low ends, that will pay you off, you should bet and re-raise to suck all the money out of them.
Now if you have a high hand the plan changes.
If you are against several low hands, and it is certain that the pot will be split, you should just check and call if you know that the lows haven't improved into flushes or straights.
You are doing this because you are hoping to split the pot.
If you are against a couple players who seem like they would call with weaker high hands, then go and bet.
Here is a common situation.
Let's say you are going for low and you missed it on the river but you paired a card.
You are against a player that has been betting his low throughout the hand.
If it isn't obvious that he has a straight, flush or aces, you should call, this is because he might just have the better low hand or a botched low with a small pair.
Your pair might be able to get you at least half of the pot.