If you have an issue with Wifi when using the Samsung Galaxy Mini then there is a few things you could check.
When you want to browse the internet you could use 3G or Wifi. 3G will cost you money as you will use your network data usage, where as Wifi can often be free because you are using it through the home network
However if you could not connect your device to the Wifi network then you have a problem. Here's a few things to do:
Connect the device to the Wifi network using IP address
1.Go to Apps.
2. Select Settings.
3. Select Wireless and network.
4. Then choose Wi-Fi settings.
5. Press Menu key.
6. Select Advance.
7. Select Static IP.
8. Now put in the the Wifi network IP address, gateway, netmask, dns1, dns2.
To find out this information you can it from the network router or from one of your client PC.
Reboot the device
You could reboot the device and see it works. Take out the battery for 10 seconds and put it back in.
Dial a code
Dial in this code *#*#526#*#*.
Check Wifi Encryption Type
There are several encryption type such as AES or TKIP ensure you have use the correct encryption to connect to your router modem.
Reset the Device
As a last resort you could reset the device and see if it works. Samsung Galaxy Mini: How to Hard Reset.