Of course there are no skill checks in most OSR games. So how do you decide whether your guys can do a particular hard task?
0. Easy Tasks
By golly, don't roll for easy tasks! These guys are heroes and they shouldn't have their adventure derailed because nobody can open the jar of mayonnaise or something.
Even if it comes to a mountain climbing expedition, you probably shouldn't roll for normal conditions with the right gear. They're adventurers, and they should be able to do adventures!
But what about for hard tasks?
1. Stat Checks
If a particular task is related to a particular ability score, then it follows that you can use a check against that ability for success.
Roll a d20 and if that result is lower than or equal to the stat in question, it's a success. In this case, even some poor sod with a 3 still has a 15% chance of doing something hard. You could also check against 3d6 or 4d6 if you want to. A 3d6 would make it easier than a 4d6 because you want a low number.
This is an easy way to decide whether a hard task will succeed or fail.
Another way to do a stat check that is less dependent on the actual stat number is to roll a 1d6 and add a +1 for a high stat or a -1 for a low stat, and then see if the result is a 5 or 6. If it is, that's a success. You can adjust the range of success for an easier or harder task.
This is a good way to do it if you worry about the overall importance of stats. It's they way I do it in my game.
2. Saving Throws
Saving throws are not tied closely to stats; they are tied to class and level. Therefore if you are a higher level guy, like a Hero or something, you will just be more competent than a level 1 or 2 duffer. This is they way they did it in the olden days according to some of the vets who were there. There is evidence to support this in the Thief guy skill tables. First, the Thief gets better as he gets higher levels, and in the case of Pick Pockets and the assassin's assassination tables, the chance of success depends on the level of the victim. This way also favors Demi-Men because they get bonuses to their saves.
I think we will try this way and see if we like it better.
3. Hybrid Methods
You could use Method 1 for actions and Method 2 for reactions. I think they do that in ACKs. Or you could use Saves only but modify them with stat modifiers. These seem unnecessarily clunky to me but maybe it's just right for you.
4. Something Else
What do you guys do when you have to check for success?