It's important to buy from a reputable dealer who has a "no questions asked" return policy. Remember, unless you are purchasing a synthetic, mass-produced drum, all drums made of wood and goat skin have a distinct voice. It's ok to purchase a drum on the internet as long as they provide clear contact information and have a good return policy. As with many other products, there are plenty of people out there buying large quantities of cheap drums and dumping them into the market.
Ashikos are constructed using two different techniques. They can be carved from a single piece of wood or glued up from staves or strips of wood. If they are carved from a single piece of wood they have a greater tendency to crack or split. World Wide Drums sells Ashikos which are constructed from glued staves of wood. This technique produces an Ashiko that sounds great and does not crack. There is nothing wrong with carved Ashikos. Just beware of the issues with potential cracking.
Some Ashikos come with fur on the playing surface and others have the head shaved. Drums with fur on the playing surface produce a more muted or muffled sound. Some people refer to this as a dry or dead sound. Ashikos with a shaved head are more open sounding and tend to be louder. Fur on the playing surface is a matter of preference. I prefer to have the head shaved. I find this to be more versatile. You can produce both open and muted tones by changing your technique. For an open tone, bounce your hand away from the head and "pull" the tone from the drum. For a muted tone, either rest one hand on the head while paying with the other hand or leave your playing hand, i.e. the hand making the stroke, on the drum after the stroke long enough to mute the tone.
Most Ashikos have the fur left around the rim of the drum regardless of whether the playing surface is shaved. This does not affect the sound. It is there more for esthetics. Some people find it more comfortable to play a drum with fur on the edges. The fur provides some cushioning for your hands as you play slap beats.
Look for drums laced with Alpine rope and not just heavy string.
What should you listen for? Listen for open, deep bass tones and a nice sharp crack when slapped. Look for a drum that's easy to play. You should not have to use a full force blow to extract a nice tone. The drum should sound good or respond at all volume levels. The tone should have a nice sustain without strange overtones or pitch shifts/bends. If you do encounter these symptoms, they can sometimes be corrected with tuning. They can also be the result of a poor quality head or an uneven bearing edge. If it's just a tuning issue, you can sometimes test this by placing the drum on the floor, holding it down with one hand, striking the drum and pulling up on the lacing. If the pitch evens out or improves, it's likely just out of tune. Another trick is to place a heating pad on the goat skin drum head at low setting for a few minutes. This may bring the pitch up enough to take out the overtone.