While I strongly prefer the ease of a class and level system of character development over a skill-based one, there are times when a player rightly expects that something in the character's background should apply to a task or problem in an adventure. Here's a quick and dirty system that has roots in an already existing B/X rule.
Whatever the character did before he or she became an adventurer is his or her skill set. The default assumption is that the PC possesses these skills at an apprentice level. In game terms, this means that the PC has a 2 in 6 chance to succeed at tasks related to these skills - in exactly the same way that a dwarf character has a 2 in 6 chance to detect certain features of mines and caves, because all dwarves are taught basic mining skills.
This chance is for things that, in the judgment of the DM, are so difficult that an ordinary layperson would have little hope of success. In the interest of not discouraging players from attempting actions simply because their characters don't have a particular skill, the layperson should succeed on a roll of 1 on 1d6.) A mere apprentice isn't a sure thing, either, but he's still twice as good as the layperson.
Some actions may be so basic that even a person with no special training can succeed without much difficulty. For example, a character with a background as a pearl diver might be a better swimmer than everyone else in the party, but anyone ought to be able to swim unencumbered in a calm lake without need of a die roll. On the other hand, maybe the DM deems retrieving an object from 30 feet of water in choppy seas to be a difficult task. Any character may attempt it, but the pearl diver has the best chance of success.
If an obstacle or task would ordinarily require a roll which affords an unskilled character a good chance of success, then the DM may rule that a character with the relevant expertise succeeds automatically. For example, if the dive above allowed a 3 in 6 chance for any fool to succeed, then the pearl diver might succeed without a roll - it's routine stuff for him.
Very rarely, if ever, should a d6 skill roll determine a character's death or survival. In the example above of the dive, either the character succeeds or comes up gasping for air, having failed to retrieve the item. Of course, other things could happen while the character is underwater - a shark could attack, or he could get his foot trapped in a giant clam - but a failed skill roll, in and of itself, should rarely be lethal, and then only with advance warning from the DM that the task is so hazardous.
Most characters should start with one trade, i.e. one broad skill set. Dwarves may choose one other trade in addition to mining (or at the DM's option may double up on mining and start as journeymen with a 3 in 6 skill roll.) Additional skills may be learned, and old ones improved, at the DM's discretion. This isn't something that can be done during a week off beween adventures; it should require either a long period of down time (at least several months) or extensive use of skills during adventures. For instance, if the party advances a couple levels in a series of maritime adventures, it's reasonable to allow them to gain an apprentice's proficiency in seamanship - provided, of course, that they're actively assisting the captain and sailors in piloting the vessel. If the party magic-user stays below decks studying history books, he's not going to learn much about ships and the sea.
Gaining greater proficiency bumps the chance of success up by 1 point, i.e. to 3 in 6. This should require that the character spend an even greater amount of time learning and practicing than is required for apprentice-level skill, or that it be central to his or her adventures for at least three or four levels of experience. Increasing to 4 in 6 or better should really be beyond the scope of an adventurer; this level of skill is the province of people who make it their lifelong vocation, not vagabonds and thrill-seekers who dabble in it on the side.