This glitch has been tested and will not corrput any previous game data. This glitch works for Pokemon Red and Blue.
Make sure you can/have/are able to:
1. Have a pokemon who knows the HM move FLY
2. Able to access Route 8 (East of Lavender Town)
3. You HAVE NOT defeated the trainer north of Cerulean Cith who has a Slowpoke
4. Quite a few Ultra/Great Balls or a Master Ball (Prefered for a quick snatch/steal)
To encounter a Mew, you MUST NOT defeat the Youngster with the Slowpoke north of Cerulean City (towards Bill's Place). Skip this trainer and continue with your adventure until you are able to access Route 8 which is East of Lavender Town. There, the most recognizable landmark is the row of four trainers facing which way. It you venture farther east, you will find a gambler who is distinguished by his black body and egg shaped head. Just to get a visual of his surroundings, look around him, but DO NOT get caught by him. If you do, you have messed up your chance.
After scouting out the surroundings of the gambler, fly to the other entrance of the Underground path way (I believe it's west of Celadon?) and come out the other side just to be outside the door (I just walked around the Gambler, but it was kinda tricky). DO NOT STEP OUT AT ALL FROM THE DOOR because here comes the tricky part. Get ready. Take one step out of the doorway and IMMEDIATELY press PAUSE. It will bring up the menu. Then, use your pokemon you should have that knows fly. Fly to Cerulean City. After your pokemon has the orders to, the Gamblers will notice you, but he will not get the chance to approach you. If he approaches you, you've done something wrong.
At Cerulean City, venture north to meet up with the Slowpoke kid. Battle him and win (Which should be easy duh). After you've beaten him, don't take another step. Fly directly to Lavender Town. There, travel east towards Route 8 again. Once you've reached the small opening between the town and route, your menu should suddenly pop up. If it does, do not react. This is a good thing. Simply exit out of your menu, and automatically, you will encounter Mew. It will be at Lv 7, but it will be tough trying to catch it.
Have fun Catching 'em.