Which Mac OS are you running?
What sort of internet connection does your home have, Cable or DSL/ADSL?
My guess is your running on DSL with a PPPoE from your modem into your wireless router and its not "always on", it just connects to the internet when required, so even thought your home network is working fine (ie full bars) your internet connection only works when you open Safari.
You will need to adjust the PPPoE settings of your Internet connection, not home network, so your internet connection is always on.