There are 151 Pokémon in Japanese Blue's Pokédex. Only 150 are needed to get the certificate from the Game Freak employee in Celadon City. The game is programmed to accept any 150, meaning if you use the Mew glitch to get Mew, you can skip another Pokémon. That said, this page focuses on getting the 150 Pokémon that aren't Mew.
In order to complete the Pokédex in Pokémon Blue with only Gen 1 games you will need access to at minimum a copy of Pokémon Red and a copy of Pokémon Green.
Access to only Yellow is not sufficient, as Ekans, Arbok, and Electabuzz are only available in Red, while Magmar is only available in Green.
The following version-exclusive Pokémon have to be traded from another game:
Several Pokémon are mutually exclusive, meaning that if you get one species, you cannot get at least one other species. The list of mutually exclusive species is below.
If you have access to Pokémon Stadium, you can get all of these by defeating the Elite Four and Rival.
Otherwise, you will still need access to two games anyway to finish the Pokédex. So, you can have each game choose a different Starter and Eeveelution, and then
trade around to get all the Pokémon registered. Make sure that you trade each Starter and Fossil at each evolutionary stage, since you can't un-evolve Pokémon.
If you have access to Yellow Version, you can also just trade all the Starters from Yellow, because Yellow players get all three Starters.
Four Pokémon evolve by trading: Machoke, Graveler, Kadabra, and Haunter. You can get Gengar and Golem via in-game trade, meaning you will only need to trade for Machamp and Alakazam. The easiest way to get data for them is to trade them to another game and immediately trade the newly-evolved Pokémon back.