Travertine can be stained, dyed and enhanced, but there are limitations.
Traditionally, natural stones cannot be stained due to the inherent durability of the stone and its inability to soak up moisture of any kind. Slate and granite, for example, are impossible to stain. Travertines and marbles, on the other hand, are softer stones that readily soak up liquids, which means you can alter the color with stains based on your personal preference.
Stone Enhancers
If you are looking to enhance the natural color of the stone and bring out its richness to showcase a darker side of the stone, you are better off using a stone enhancer rather than a stain or ink. Travertine is always lighter in color prior to a stone enhancer being used on the surface, and many homeowners mistakenly use stains and dyes before they check to see if the color they are looking for can be brought out by a coat of enhancer.
Wood Dyes
Wood dyes can be used to stain travertine, although they can only change the color of the stone in shades. You should always buy a stain that is one or two shades lighter than the actual color you are going for, and apply the stain in coats to see how the stone is going to react to the color additions. You can always add an extra coat later, but you cannot regress once you have darkened the stone.
Inks
Inks are reserved for the darkest types of coloring. While wood dyes can be used to darken the stone's natural colors, inks can go one step further and actually alter the overall color of the stone if it is a light-colored stone to begin with. Similar to using wood dye, you should always buy your ink slightly lighter in shade than your final choice and apply it in coats until you achieve the results you are looking for.
Limitations
Travertine is still a natural stone, which means it can never completely change its base colors unless you are dealing with a light-colored stone. Tans and earthy-toned travertines are impossible to stain completely, and all you can do is darken the natural tones of the stone, depending on what type of dye or stain you use during the coloring process. Always test a small area or a test piece of stone beforehand to get a feel for how your particular stone is going to be affected by the stain or dye.