OK, first of all, the Governors should be aware already, especially the link governor for safeguarding and the Chair of the Board of Governors (the terminology may vary due to whether or not this is an academy, a single academy, a multiple academy trust, local authority maintained or church school - you'll be able to find out on the website about the Governing Body structure). However, this cannot be guaranteed, so the best way is to a) notify the Head that you are making a formal complaint about the staff member's conduct and b) you wish this to go to the Governing Body. If you are unconvinced whether this will actually get there, the contact details for the clerk to the governors is public information that should be listed on the school website (along with the complaints policy) and you can contact them directly - even if it needs to be referred back to the school at first, once it's been received, it can't be hidden.
When you look at the policy, it should give a timescale for when a meeting is arranged (eg, 15 days from receipt of your complaint) and whom it should be with - the resolution manager is the official title, and could include the Head as the RM for a Stage 1/2 complaint as an ex officio member of the Governing Body, but might not - it really does depend upon the complaints policy. There should then be a panel/meeting convened as soon as possible, which will be fully minuted by the Clerk, who is very well aware of the importance of accurate recording and impartiality - if you confirm your availability for a face to face meeting at the outset (3 dates/general times), it'll make it easier for the clerk to get sufficient governors together. Don't agree to meetings without somebody minuting them.
In addition to this, there is also Ofsted and the Local Authority, as this could trigger a zero notice Safeguarding Inspection. As it's likely that the DSL has literally only just filed the safeguarding audit with the LA in the last month and it might not have even been advised to Governors yet if they haven't yet had their full governing body meeting, that in itself will trigger some very pointed questions, both from the LA and, if they're a decent board, from governors.
They may be reluctant to say or do much at first because it should be the subject of a disciplinary at the same time as the criminal investigation, but don't let that stop you, any more than an open 'that's appalling and I will act right now' shouldn't. This isn't going to go away.
The DSL should be dealing with the offence/safety side of this. That is separate to the complaint about the twat teacher. You need to be absolutely clear that they are two separate things which can be complained about separately - it's normal in some circumstances to have two complaints running concurrently regarding complex matters.
The main thing is to not let this fizzle out with any 'he misunderstood/she misunderstood what he meant, he was just concerned for her safety drinking' or 'he's really, really sorry and will be having further training' responses.