Assuming the event items (Old Sea Map for Faraway Island and AuroraTicket for Birth Island) were obtained legitimately, the following conditions need to be met before the player can access either island.
Emerald
1. Reach Lilycove City
2. Present the Old Sea Map to the attendant at Lilycove City Harbor
That's it! Mr. Briney will take you from the harbor as long as you have the Old Sea Map in your bag. Faraway Island cannot be accessed in any other game.
Emerald
1. Obtain the S.S. Ticket
2. Present the AuroraTicket to the attendant at Lilycove City Harbor
The S.S. Ticket is received from Norman after entering the Hall of Fame, meaning Birth Island is a more involved journey than Faraway Island.
FireRed / LeafGreen
1. Obtain the Tri-Pass
2. Obtain the Rainbow Pass
3. Present the AuroraTicket to the Seagallop in Vermilion City
As the Rainbow Pass is only obtainable once the player has the National Pokedex, the FRLG process is more tedious than the others. It also involves clearing a significant portion of the Sevii Islands story, so brace yourself for a long journey.