If they have even the slightest doubt about your intentions they will hold back.
So whatever you would normally do to indicate your interest to a confident man, double or triple it for a shy man (in proportion to his shyness): While I don't suggest that you overtly ask out a shy guy on an "official" date (this would too flagrantly undermine what he knows to be his role as described above), I do suggest that you suggest and then ask him to hang out under casual pretenses.
A shy man may never become courageous enough to approach women in general, but he can eventually grow comfortable with a girlfriend or wife.
As the fear of the unknown dissipates with increased familiarity, a shy man's confidence in his interactions with you will rise.
Unfortunately there is no quantifiable or simple rule here, as his threshold for embarrassment will vary in proportion to his shyness and his pride, which are different in every man. Well, for starters, in the early stages, you will need to give him blatant signs of your interest.
Aside from their innate male tendency to overlook anything that isn't spelled out in black and white, shy men will by default doubt any sign of your affection.
If a girl doesn't approach or initiate contact with a shy man, nothing will happen.
Dating a shy man takes a lot of initial investment for an uncertain outcome.
That offense will reflect negatively on you and the relationship.
So be careful about how much (and which) initiative you take.
Of course, he will question this over and over in his mind, wondering whether you were asking him to hang out just to be friends, or if you wanted something more.
So show him that you want something more - ask repeatedly.Once this happens, you will need to gradually assume a less assertive (more traditionally female) role, because he will then be more liable to get bored with you, wonder if he could do better, consider his other options (the perception of which his newly-found confidence will likely inflate), fool around, etc.