Caring for a Damascus knife is simple, and how much it asks of you depends on one thing: whether the steel is carbon or stainless. Carbon Damascus needs drying after use and a little oil now and then. Stainless Damascus needs only ordinary cleaning. The pattern itself needs nothing special, because it runs through the steel and cannot wear off. Here is the whole routine.
The one rule for any Damascus knife: dry it
Water is the only real enemy. Whichever steel your knife is, wash it by hand with warm water and a little soap, then dry it right away with a towel. Do not leave it wet in the sink, do not let it drip dry, and keep it out of the dishwasher, where heat, long soaking, and harsh detergent are hard on any knife. That one habit, drying it, covers most of what a Damascus knife needs.
Caring for carbon Damascus (DSC® Carbon)
Carbon Damascus (1.2842 + 1.2767) is reactive, the way all carbon knife steel is. Beyond drying it, give it a thin wipe of oil when it will sit unused for a while, or if you live somewhere humid. Food-grade mineral oil is the standard choice for a kitchen knife, because it is safe for food contact and does not go rancid. A camphor or knife oil is fine for a knife that stays out of the kitchen. Wipe a few drops over the blade with a cloth, and wipe off the excess.
If you cut something acidic, a tomato, citrus, an onion, rinse and dry the blade when you are done rather than leaving the acid on it. None of this is fussy once it is a habit.
Caring for stainless Damascus (DSC® Inox)
Stainless Damascus (1.4034 + 19C27) is far more forgiving. Its chromium resists corrosion, so once the blade is hardened and finished, it stays bright with ordinary washing and drying, and it does not need oiling to prevent rust. This is why DSC® Inox is the easy choice for a kitchen knife or a knife that someone else will own and not think about. It is also approved for direct contact with food. There is more on that in Is Damascus steel food safe?
A patina is not rust
On a carbon blade, a gray or blue-gray film will build up with use. That is a patina, a stable oxide layer, and it is a good thing: it is food safe, it protects the steel underneath, and it deepens the look of the pattern. Leave it be, or encourage it. Rust is different: it is orange, it flakes, and it comes from leaving the blade wet. If a few rust spots appear, they lift off easily with a paste of baking soda and water, or a fine non-scratch pad, then dry and oil the blade. For the full picture on carbon versus stainless and moisture, see Does Damascus steel rust?
What to avoid
The dishwasher, for any knife.
Soaking, or leaving the blade wet in the sink.
Cutting on glass, stone, or steel, which dulls and chips the edge. Use wood or a soft board.
Storing a damp knife in a closed drawer or sheath, where trapped moisture invites rust. Store it dry.
The bottom line
Dry your Damascus knife after every use and it will last for decades. Carbon (DSC® Carbon) wants that plus an occasional wipe of oil and welcomes a patina. Stainless (DSC® Inox) wants little more than washing and drying. The pattern takes care of itself.