Alcohol and water or detergent solution will not harm any video card so long as the fan is removed.
There are many ways to clean a card, consider that after manufactured with today's water based fluxes, they clean them with hot detergent solution. Therefore, if the stickiness you see remaining on the card is some flux (I see this VERY often, the flux residue looks pretty transparent but after some time it turns a cloudy light tan color), use hot detergent solution and a toothbrush to scrub it away. Take the heatsink/fan off the card first, meaning you will need heatsink grease to put it back on after cleaning off the original thermal interface material.
For that matter, I've even washed fans themselves in hot detergent solution, but don't submerge them and don't make the solution very strong because you want to minimize, if not completely eliminate, any detergent getting inside the fan bearing. I suggest not cleaning a fan like this unless it is absolutely necessary, usually it is enough to take a paint brush and wipe the dust off.
There is nothing wrong with using alcohol, or contact cleaner, but be aware that these may dissolve the ink on the stickers affixed to the card, but for that matter getting the stickers wet will make them look bad too and either way it might effect whether the card is still under warranty. Do not use more aggressive solvents like acetone, as it may dissolve the plastic used on some heatsink mounting parts, fan assemblies, or covers.
Alcohol is better used to clean off the residue from soldering back when they used rosin based flux. It or an ammonia based cleaner like Mr. Clean are better than detergent solution alone if you have tar residue, like from an industrial environment or a heavy tobacco smoker. The bulk of the goo will come off by applying it and letting it sit with a paper towel over it then rinsing, but to get it really clean you'll need to use a tooth-brush to scrub the area.
If whatever you use leaves a residue, like goo-gone, or detergent solution, finish up by thoroughly rinsing that off. Plain tap water will suffice if it's a detergent solution but some other cleaners need an alcohol rinse, or an intermediate step of washing with detergent solution then a plain water rinse. After rinsing with plain water you must wait a long time for water under the chips to dry if you don't bake the card or point a fan at it. Even pointing a fan at one I wait an entire day after shaking excess water out. To speed up the drying time you can do a final rinse with pure alcohol, or even the 70% alcohol you find in drugstores, it will dry faster and is less conductive.
There is nothing special needed to wipe the card. Lint all over it is not a problem... you'd simply need to wipe the contacts when finished so they are clean. After all, the moment you put it in a system again there will be dust getting on it anyway. A regular paper towel does fine to wipe the contacts, it is not a biological experiment where a tiny piece of paper towel lint matters, the slot contacts will push those out of the way of they aren't excessive in quantity. However, after rinsing and shaking excess water (or other non-residue solvent) off the card, you don't need to manually wipe it dry. Just point a fan at it for a day.
Another reason to avoid wiping a card is that can create significant static electricity, and blowing off a card with compressed air or using a vacuum does too because of the movement of the dust particles at high speed. Often this won't be a problem, but it is far more likely to cause one than using alcohol or which type of paper products leave lint. The greatest likelihood of harm while handling or cleaning is generation of static charges.
Video cards, motherboards, memory, PSU - many different parts in a PC (besides the hard drive and optical) can get wet with no problem so long as they are completely dry before power is applied again. Just expect the stickers to look bad and the fans must be removed, heatsink grease reapplied if present. Also remember that water collects in connectors, shaking out excess and allowing ample drying time is a must.