Our responsibility to good judges is expanded in Romans 13:7.
Give everyone what you owe him: If you owe taxes, pay taxes; if revenue, then revenue; if respect, then respect; if honor, then honor.
The basic principle is that we should make payment to everyone whom we owe something. When judges do their work well, the whole of society benefits, not just the people who get their cases heard. If judges deal with crime effectively, society will be peaceful and the economy can grow. We all benefit from their work. We all owe something to the judges who make good decisions.
The word taxes is a little misleading as it implies a compulsory payment that is decided by the government receiving. A tax must be paid, regardless of whether anything has been received in return. Paul is talking here about a voluntary payment, not a compulsory levy. He is telling Christians that they should decide what they owe to the judges in their community and make sure they pay something towards their upkeep.
The Greek word translated as revenue can mean toll. Tolls are legitimate because they are payment for a service. For example, if I use a road, I owe something towards the cost of maintenance of the road. A toll is a payment for a service.
The thrust of this verse is that we must decide what we owe to whom. We only honour those who are worthy of honour. We only respect those who are worthy of respect. We are not required to respect the governing authorities, because they are usurpers of Gods authority. We do not need to honour politicians, who make human laws, because they are in rebellion against God. We should honour and respect good judges, who apply Gods law wisely.
We should only pay money to judges that we trust. Christians should never subsidise evil.