Use a larger pot
Unfortunately, the plastic container in which the basil plant is placed does not do much good for the life of the plant. Throw it away and put the plant in a larger pot. Preferably a pot with holes at the bottom so that you can put it in a layer of water.
Looking for a nice spot for the basil? In this video, Marianne shows you how to use a wall rack as a plant rack, among other things.
Let the sun shine
If there is one thing your basil plant cannot live without, it is sunlight. So place it by the window or even put it outside (it does very well here, too), so it gets plenty of hours of sunshine every day.
Water, water, water
Skipping a day is not a total disaster, but if you really want your basil plant to stay alive for a long time, you will have to water it every day. Do not water the stems, but pour water into the saucer underneath the pot (hence the pot with holes). This way the roots absorb the water better.