Interesting one. I think there are three aspects to it, and I am not quite clear which one is more important to you, so here goes !
At the simple level, if this person merely wishes to access the net through your router, then they have to apply the same security protocols to their computer as you have set on your router. This is now normally WPA-PSK2. They also need your wi-fi access code - the WPA-PSK2 key - to access your router.
Security considerations also apply to his computer and its internet security. At a mininum, you would need to assure yourself that he/she has current AV and anti-malware protection, which is kept up to date. That would be considered a reasonable step. As a horrible example, some years ago one of my neighbours had his computer taken over as part of a botnet. His ISP sent him an e-mail terminating his contract immediately for breach of T&Cs - it had been generating spam. It took him some effort to convince the ISP that he was an innocent victim ( if a careless one ) and he had to prove he had adequate security before they opened a new acount for him.
Security of your files and data is another concern. This can be done by not sharing files etc. on the network. It is entirely possible to set up a network so that computers 1 and 2 can share data, but not computer 3.
If you are referring to something like BT's 'Open Zone' which turns every BT router into a wi-fi hotspot, then this does not, as far as I am aware, create any problems or responsibility for you. To access the internet through this system, you need an access code which the user obtains from BT and NOT from you.
So, to sum up. If your friend is wanting to share your router, then you need to take some precautions. If they merely want to use some semi-public hotspot that happens to toriginate from your router, then you have no responsibility.