There is no way to change the odds above the standard 1/8192 chance in Emerald, and on top of that, Emerald's RNG is glitched with a fixed seed, which means that every time you start the game, it'll reroll the same frames in the same order. This doesn't really have an affect on random encounters, but it does make SRing almost impossible because you'll keep hitting the same 20 or so frames. There are some workarounds, though.
For all the legendaries in Emerald, you'll want to use the run away method; encounter it, run away if it's not shiny, leave the room and reenter, and the legendary will be back. By doing this, you can encounter the legendary as many times as you need without turning your game off and will get a unique roll every time.
This method won't work for the starters, though. In order to get a shiny starter, you'll need an early shiny frame, which requires a lucky TID/SID combo. To get this, just SR like normal, except every 60 SRs or so, create a new save file to scramble your TID/SID combo. If you get a shiny Zigzagoon before you get a shiny starter, it means you've found an early shiny frame and should stop creating new save files; as long as you use the current file, it shouldn't take you too many more resets to find your starter.