Why do I need to pay connection costs for Portbase?

Connection costs are charged because a number of steps must be completed in order to make use of the ‘digital border’ of Portbase.
Portbase is a cost-effective organisation. The connection costs are € 249.50 per service.

Does the collaboration relate to the process on the UK side as well?

No; you can read more about that here.

Will Customs perform fewer inspections because of the port solution?

Customs performs risk-based inspections. As part of this, it also verifies whether the correct formalities have been completed (e.g. the presence of an export declaration, otherwise the shipment is prohibited from leaving).
The solution will consequently not lead to a reduction in the number of inspections. The starting point of Dutch Customs is smart supervision with minimum disruption to the logistics chain, under the assumption that all formalities have been completed. The process that has been automated in the ports via Get Ready for Brexit makes this possible.