Art is important for a game to do well, perhaps more than most indie developers want to admit. Successful games, even those without many art assets, have a level of polish to them - something we've come to subconsciously expect from all games, even though the rational part of our heads.
Many programmers just throw some programmer art (and it's almost always poorly-drawn cartoonish sprites, often made in MS Paint) into their prototypes/alphas and call it a day. Not that there's anything wrong with that as such - art is *hard*, and demands a great deal of dedication and practice, something that many people don't have the aptitude or inclination for.
What it means, though, is that the project is held back. If it looks unpolished, often it will *feel* unpolished (whether the two do more than correlate is an interesting but volatile debate). It's important to get your art assets in as early as possible.
Of course, that's easier said than done. In addition to art being hard to make (especially for most programmers who may not have the skill), it's also time consuming - time that could otherwise be spent improving the code, or adding features, or fixing bugs. And on top of that, sometimes the game's feature list will change during development, meaning some assets become redundant or new ones are required, or both.
Whether or not you manage to get the final assets into your game quickly, you nevertheless need to put some thought into how to make your alpha builds usable and visually appealing enough that playtesting doesn't want to make you vomit. Having some idea as to the art style is important too, if it's not already part of your design document or planning, as art style informs and is informed by tone, which can be altered and communicated through mechanics and writing just as much as it can through the art.
For CivClicker II, the goal is to make a game that's colourful and visually appealing, communicating a more positive tone than the greys and browns of a modern military shooter. Ideally the art should be simple and iconic - like the game's mechanics, which should ideally be as straightforward and transparent as possible while still providing depth and interacting to form deeper complexity.
Later (hopefully not too much later) I'm going to be looking to replace the current art map with something that looks less like I drew this using MS Paint. Fortunately the map lends itself well to tilesets, and the game's heritage means I can get away with a mixture of clean lines and pixel art. Ideally I'm going to need to keep things clean enough to communicate other information on the map beyond the tiles, so there's a deeper need for simplicity there than simply keeping the game user friendly. Lastly, the game is HTML5, so I may well be able to use some of the more interesting properties of the canvas element later, as well as using CSS for the interface (which I am vastly more experienced in).
( The fact that a simple and iconic art style also allows me to re-use assets from the original game is neither here nor there... ;) )