When I started playing SW-TOR, I followed the thought of wanting my alts to cover all of the crew skills, but limiting my choices to gathering skills that would support my crafting skill on a per-character basis. (E.g. for a character with a Cybertech crafting skill, I would take Scavenging and Underworld Trading.)
The aforementioned approach worked out all right for me, but, with Strongholds and guilds and the ease of transferring items from character to character, if I started all again, I would have chosen my Crew Skills slightly differently, instead trying to assign the Crew Skills to the classes that would use them best and transferring materials as the other crafting skills would require. (I.e. I would have my character with Scavenging transfer the materials to my other character with Cybertech.)
For each Crew Skill, the best classes are (I consider Critical chance as always better than Efficiency):
By "Droid" in this list, I mean the Ship Droid (C2-N2, 2V-R8) with a specific sensor.
- Bioanalysis: Sith Warrior, Smuggler, Bounty Hunter
- Scavenging: Trooper
- Slicing: Sith Inquisitor, Jedi Knight, Imperial Agent
- Archaeology: Sith Inquisitor
- Diplomacy: Imperial Agent
- Underworld Trading: Sith Inquisitor, Imperial Agent, Droid
- Treasure Hunting: Sith Warrior
- Investigation: Sith Warrior, Smuggler
- Armormech: Trooper
- Armstech: Smuggler
- Artifice: Sith Inquisitor (Efficiency only; Jedi Knight has a smaller Efficiency bonus)
- Cybertech: Droid (Sith Warrior, Jedi Consular, Imperial Agent, Smuggler all have +10 Efficiency bonuses)
- Biochem: Jedi Knight
- Synthweaving: Jedi Knight, Sith Warrior, Droid
As a result of this data, if you choose to have one character of each class, I would recommend the following for Crew Skill distribution (Italics indicate that the class doesn't have the highest rating in the particular skill):
- Sith Warrior: Synthweaving, Bioanalysis, Treasure Hunting
- Sith Inquisitor: Artifice, Archaeology, Underworld Trading
- Bounty Hunter: Bioanalysis, Biochem, Treasure Hunting
- Imperial Agent: Slicing, Diplomacy, Cybertech
- Jedi Knight: Biochem, Slicing, Treasure Hunting
- Jedi Consular: Cybertech, Scavenging, Diplomacy
- Trooper: Armormech, Scavenging, Bioanalysis
- Smuggler: Armstech, Bioanalysis, Investigation
For the classes where I indicated non-optimal (in Italics) skills above, I generally selected skills in which the class had some bonus, but for most characters, I would just default to Biochem, Bioanalysis, and Scavenging.
A note about Diplomacy: SW-TOR divides the Diplomacy missions along Republic/Empire factions, but does not distribute them very well. So, even though the Imperial Agent has the best companion for Diplomacy, you may still want to select a Republic class to have Diplomacy as well (the Trooper and Jedi Consular both have companions with Critical bonuses in Diplomacy, as well as the Ship Droid with the Medical Sensor).