As a general rule, you can use open-source code under one of these licenses within a website without any issue. This applies even if you are charging to implement it or charging to use the site. The license only becomes an issue if you develop a program that you wish to sell that includes / uses code that is available under one of these licenses. Then you have to check the wording more carefully.
I'm not giving you a legal opinion so if you feel that you need one, then go for it. I looked into this a while back because I am using some GPL code in a number of systems that I built. These systems are not open-source and they are not sold or distributed. I use them as an online service. I did not get a legal opinion, I just did some research on the Internet. What I found was that using GPL code in a system that isn't distributed / sold is ok and does not require you to release all of the source and does not limit you from using the GPL code. Apparently, some people consider this to be a loophole in GPL. There are versions of the GPL that do address this "loophole" so you need to look at the specific license for the software that you plan to use to see if it has the additional provision for ASP (Application Service Provider) systems or not. You can use the link below to get a bit of background on that:
JQuery is dual licensed under the GPL and MIT licenses. From what I can see, it uses the original GPL license and that addresses "distribution" not online use as an ASP. I strongly suggest that you read the licenses referenced by JQuery for yourself. Remember that each plugin is also licensed so you will need to look at those as well. You can also read posts about the JQuery licenses here. If you still have concerns after that, you can post your own questions on the JQuery forum or ultimately you can see a lawyer if that is what you need to be able to sleep at night.