Please provide some more information: 1) Mac Mini - vintage, i.e. month/year; 2) Wi-Fi brand/model; 3) Have you tried a wired connection to the Mini, as suggested; 4) Are there other Wi-Fi devices in the house that are working w/ the Safari browser; and 5) Have you reset the settings in Safari (not sure what the options are in the older OS X)?
As suggested by Harry, you might have browser malware which Adware Medic may resolve - also 'power recycle' all of your hardware (i.e. pull the plug and/or turn off the switch) - wait a minute or so, then re-power the modem, router, and Mini in that order - post back the answers to the questions above and the results of the suggestions now made in several posts. Dave