If there's one thing I have learned over the years, it's that musicians LOVE to carry on at length over oils, cleaners, polishes, and the like... (Witness the length of this thread...)
I've been a cabinetmaker all my life, seen more wood than most people, and I am continually amazed by the misinformation that is spread around, and, since the dawn of the internet discussion forum, is spread around to a wider audience and at a more alarming rate!
Firstly, the overuse of oils of any kind on the fingerboard is possibly the worst thing! There's a commonly held folk wisdom that says that the fingerboard needs to be "fed". Wrong! The wood in the fingerboard is dead! It needs no feeding! If you continually slather your fingerboard with oil, especially the wrong types of oil, you are providing a place for dust and crud to build up. It doesn't do anything for playability or well-being of the wood, and just needs to be removed later.
All oils are not the same! Some folks recommend mineral oil or other petroleum-based oils. Worst thing you could do! Petroleum-based oils DO NOT polymerize when applied and exposed to oxygen. They simply remain oily. Better choices are plant-based oils like boiled linseed oil. which will polymerize quite nicely and get to be a relatively hard consistency. This is why boiled linseed oil is the basis of a lot of furniture oil finishes. Petroleum oils are not! (Would you even consider motor oil as a wood finish for anything....?)
Don't even get me started on "lemon oil". While this is everyone's favourite, I'm here to tell you that there is nothing lemon about most commercially-sold lemon oil! It's mostly petroleum based solvents with a little lemon smellum added. I've only seen real lemon oil once in my life. It was in a health-food store, came in a 10 ml. eye-dropper bottle, and was prohibitively expensive!
If you need to clean your fingerboard or the body of your instrument, first you need to know what you're trying to remove, and what will dissolve it. If you just want to clean up dust, use a dry rag, as a lot of people here have mentioned. If you do have a build-up of gunk on the fingerboard from finger oils, sweat, dead skin cells, try a damp rag first. You don't need to slosh a lot of water around; you just need a LITTLE water and some elbow grease. The water will evaporate quickly enough. If the gunk has a lot of oily content, you can try a petroleum solvent, SPARINGLY! Your first choice should be naptha (also known as camp stove fuel, white gas, lighter fluid, it's all the same thing) This will not harm a nitrocellulose lacquer or poly finish. Do not use it on an oil finish, since it will dissolve the oil.
For an oil finished bass, though these are rare, you can get out the steel wool to remove gunk. 0000 is a good choice; it's the finest grade made. After abrasion to remove whatever, a spot application of a suitable oil finish can sometimes be helpful, but this is best undertaken for the first time upon the advice of a pro. You simply don't need much oil, and most people over-oil...
If you need to remove a build-up of rosin, try a LITTLE alcohol on a cloth. Alcohol dissolves rosin quite nicely and will not hurt a nitro or poly finish.
Of course, if you have a nice old bass where the finish has worn or deteriorated to a cracked or worn state, take all of this with a grain of salt. Keep the instrument as clean as possible with the minimum of abuse or the use of solvents of any sort. Keep your hands clean when you play, don't spill stuff on your bass, and don't slather on potions and polishes when they aren't really needed!.
