First of all, check the web designer's own web site. If it looks okay for you, then look for a link that list the web sites created by him/her/them so far.
Visit each or most of the web sites and appart from the design look also for brocken links, empty pages, not finished work, the speed at which the web pages load in the browser window.
If you find something that is not satisfactory, do not ask the designer nothing. Just look for different web designer. When you find the one with satisfactory portfolio, describe the task and ask for detailed quotation. Best of all, find at least three designers and ask each of them.
Important long terms questions that you should consider:
* who will manage the web site including the hosting and the domain registration?
You might buy:
- web design only
- web design , hosting service and domain registration provided by the same web designer
- the above plus web site updates package - to refresh the content of the web site when appropriate or on regular basis
- additionaly web promotion services
* if the domain name and hosting are provided by the designer, make sure you have complete control on both, including the coressponding user name and password for the domain and for the hosting accaount
* if you use the hosting provided by the web designer, be careful what do you pay for. It is common that a separate web hosting account that is 6-10$/month includes a number of e-mail boxes (very often unlimited number), but it is also common that the people do not know this and actually use e-mail boxes based on free services like yahoo for example, while at the same time, they have redistered their own domains
* ask the designers, if they use CMS (content management system) and whether they are going to use such for your web site. the CMSs are available in the hosting accounts for free and need just a slight customization, so web site based on such system (or based on BLOG) shall be very cheap
* ask if the designers use templates -> cheap design too.
There are many other things to ask, but it really depends on the type of the web site and your own needs and plans about it.