Grab is pretty much always among the best approaches vs shield. If you can space an ftilt on shield, that's great, it isn't necessarily "safe" even if they won't shield grab it, and it can be used to bait things out and punish their whiff. If they do whiff. Down-tilt can do a similar job fairly well, as can uptilt, but they all have to be spaced, unless you're behind the opponent or something.
Low aerials, particularly nair (though there is a timing game you can play between ac bair and a delayed one) -> are a spacie staple, and going into a second shine or grab are generally the safest options after that. But you don't have to even do the first shine, if the first aerial was sufficiently low and/or spaced, move further from the range of OoS, or if you can't be grabbed, even shield yourself if you so desire. Aerials are basically always safer if you're moving your hurtbox out of the range of their OoS options after hit, either by drifting back, crossing up, or whatever. One of the things that's special about a move like AC bair is that drifting towards the opponent, while generally not good vs shields themselves, can make acting out of shield to hit Falco difficult. The more he moves the hitbox into range, the more likely it is to interrupt such an attempt.
Aside from just walking up and grabbing, just walking up casually and shining or directly into some of the options out of shine outlined above can work.
The best thing besides just grabbing, however, is to not throw out any attack, and be at a range where you can't be hit by any out of shield option you aren't capable of reacting to yourself. If you're close enough to make a movement to make them think you'll grab, or do any of these other options, you can punish the way they react, even if it only means by taking more stage position.