If you already have a multiple entry visa for Saudi; just go to any Waseet agent (they are handling the Saudi Embassy visa process).
You'll find them in Hawally behind the Nugra Complex (North).
You'll need to bring:
- Visa for Bahrain
- Passports of your family members.
- Pictures (blue background)
- Cash (as they don't take cards, but there is an ATM in the complex).
Make sure you specifically ask for a transit visa.
Once you arrive at the Nuwaiseeb Border, you'll have to get your family members fingerprinted and photographed.
You will also need to buy road insurance at the border (don't bother buying it in Kuwait, it is not valid for Saudi).
From Nuwaiseb out, take the first right (onto Saudi State Highway 95) and then stick to the highway and you'll be headed towards the King Fahad Causeway.
At the Bahrain border, you'll need to buy car insurance for Bahrain.
Just a warning, normally they do not give transit visas for residents but as you have a multiple entry visa into Saudi Arabia they may make an exception.
Transit visas are typically only for citizens of neighboring countries, for those residents holding permanent residency in neighboring countries, or for cargo drivers and domestic helpers.