I am going to assume for the purpose of this article that you are well skilled in web design and know what you are doing.
Ideally that should not only mean that you have a bit of design talent and can make good looking pages, can do enough programming to be able to create a website which has the functionality which your clients may be asking you for, but also a good knowledge of a few content management systems so that you can quickly and cheaply create websites that your clients can easily edit and manage themselves and that will still be able to fulfill all of the needs that they have presented you with.
If you can do that then there is no reason why you should not go into this business, but nevertheless here are a few things that you should probably think about before you start.
Pros:

1. One of the biggest advantages of starting a web design business as opposed to going into any other industry with a new company is the fact that you can set yourself up and start trading straight away and with very little costs involved.
   Of course you may need to advertise to get work coming in, but even that may not be the case.
   
2. There are lots of opportunities to upsell in this market and to get your customers to buy extra services such as hosting or SEO services.
   You do not even need to worry about providing these yourself, as there are places you can go which offer reseller services to you buy wholesale and sell retail to your customers.
   
3. If you already have the skills then it beats working for someone else!
4. This is an industry which rewards innovation, so it is always possible for new companies to break into the market and be successful.
   

Cons:

1. Although there is obviously still plenty of demand for web design, part of the market is moving away from websites to things like apps.
   If you can add app development to your portfolio then you are more likely to be successful.
   Having said that, I am sure you could find another company offering app design who will let you resell your services too, so that you can add them as part of a package to include both website and app.
   
2. There are already lots of people doing this, and in a crowded market place it can be difficult to get work
3. If you have been involved in website design and development as a hobby, then you should be aware that the vast majority of jobs will not want your creative flair and innovative ideas.
   They will want something cheap, clean and simple.
   So it might not be as much fun as you think