UK citizens traveling to Ireland with a child having no passport, what's the alternative for the ID?

The fact is that all British and Irish citizens have an unconditional right to enter each other's countries. Meaning anyone with documents proving or making it plausible that they're British/Irish must be let in.

As such, if the child is yours, their birth certificate stating your relationship is enough, although some check-in handling agents (if flying) may not accept it (contact them beforehand and ask)

If it's not your child, ask the parents to obtain the child's birth certificate as well as a legalised/notarised affidavit of consent containing a copy of their ID as well as their contact details.

Be careful though. Ryanair won't accept anything other than a passport between Ireland and the UK. As far as I am aware, they are the only carrier with this requirement. I am not clear how this affects accompanied minors flying on this route with Ryanair, though.

Note also, based on experience travelling using only a driver's license, that you may be asked to produce your boarding card as evidence of your journey, this has happened to me several times recently, when entering Ireland, so be sure to keep this handy.

RYANAIR ONLY ACCEPT PASSPORTS.

I've just done a live chat to see if I can take my 10 month old baby from the UK to dublin with a birth certificate and they have said they will only accept passports.

From my own experience as both a bus passenger and travelling via private car as a passenger, the experience differs . My ID was always checked before boarding at Pembroke Dock as a bus passenger and randomly as a car passenger at Rosslare. It seems to depend on the mood of the staff and how busy they are. I have not got a passport yet , but no doubt will need to get one sooner or later. Have never been asked for ID for our child of whom we have Special Guardianship.

Where in the UK are you?

If you travel by boat, you won't have your ID checked, so another option is to drive, or Rail and Sail.

Assuming you don't have any documents in your possession other than your own photo ID, one option is to fly to North Ireland first and then take the bus/drive to Ireland. Even Ryanair doesn't require minors to have any documentation on this route and UK citizens are not obligated to carry any ID whatsoever when traveling to Ireland:

For journeys on the island of Ireland, British and Irish citizens do not require any travel documents when crossing the land border.

On the way back you can likewise take the bus/drive back to Northern Ireland and then fly home. Not the most convenient option but one with minimum documentation requirements.