The Official Shiny Pokemon Guide: Masuda Method, a Fishing Rod, and YOU! v1
In order to maximize your chances of pulling off an effective fishing streak, there are certain things to consider before heading off.
1) Location, location, location.
Before beating the fisherman's streak of 7, he gives you a hint as to the best places to go to pull off a successful streak. He mentions that areas with objects surrounding the fishing tile, such as rocks, will help increase the odds of fishing up a Pokemon from a tile. Look for small nooks and cornered-in sections of water to cast your rod in to help increase the odds of hooking a Pokemon with each reel. The more sides of the fishing tile that are blocked off, the better.
A personal favorite is a location tucked underneath Ambrette Town, seen here:
http://i.imgur.com/W0rSMTH.jpg (picture courtesy of prguitarman)
As you can see in the above image, the tile directly in front of the player is surrounded on all four sides - by the shore to the north, by un-surfable waters to the south, by rocks to the east, and by the player to the west. Searching for these types of cornered-in sections to fish in seems to result in much higher chances of reeling in a Pokemon with each cast.
*Note: It is unconfirmed how much this helps, if at all. It is being based completely off a line of text from the fisherman NPC. But play testing thus far seems to suggest it does greatly increase the odds.
*Note 2: Even though it seems to help, it is not REQUIRED to find these types of spots to fish in. I've personally fished up plenty of shiny Pokemon just standing and casting my line from the shore. With a Suction Cup/Sticky Hold lead, it should be more than enough to maintain a consistent streak.
2) The best rod for the job
If you plan on doing this early on in the game, the earliest you can possibly try is after first obtaining the Old Rod in Ambrette Town. If you're looking to hunt for other Pokemon, you'll have to wait until you receive the improved rods throughout the game.
Good Rod: Courmarine City
Super Rod: Fishing Shack, Route 16 (the very same place where the fishing streak NPC resides)
3) Field Abilities
Certain Pokemon abilities have secondary field effects that trigger and remain constantly in play while on the field. However, these effects only trigger when the Pokemon with the ability is at the head of your party. Note that the Pokemon *can* be fainted in your lead slot, and still have its ability in effect. Here are some of the most useful abilities when attempting a Fishing Streak for shiny Pokemon, as well as the Pokemon that have them:
Suction Cups - Increases the odds of hooking a wild Pokemon while fishing.
Pokemon: Octillery, Lileep, Cradily, Inkay, Malamar
Note: Excellent for the purposes of fishing streaks, as you'll almost always get a bite.
Sticky Hold - Increases the odds of hooking a wild Pokemon while fishing.
Pokemon: Gulpin, Swalot, Trubbish, Grimer, Muk, Shellos, Gastrodon, Accelgor
Note: Same as above, just with a different distribution.
Synchronize - Has a 50% chance of finding wild Pokemon with the same nature as the user.
Pokemon: Abra, Kadabra, Alakazam, Natu, Xatu, Espeon, Umbreon, Ralts, Kirlia, Gardevoir, Munna, Musharna, Elgyem, Beheeyem, Mew
Note: Very useful if you plan on trying to capture a specific shiny with the right nature. However, your odds of maintaining a streak are less reliable.
Static - Increases the odds of encountering Electric-type Pokemon
Pokemon: Pichu, Pikachu, Raichu, Voltorb, Electrode, Elekid, Electabuzz, Electivire, Mareep, Flaaffy, Ampharos, Electrike, Manectric, Emolga, Stunfisk
Note: The one and only reason to be using this over anything else is if you're dead set on encountering a shiny Chinchou or Lanturn.
Advice is what we ask for when we already know the answer but wish we didnt.