My DH and I have been married for nearly 4 years and started trying for a baby at the start of the year. We were extremely lucky and I'm due in September with twins!! 
I'm an only child, both my parents are deceased. My DH is from a large family who live inthe south east, but he's always been a bit of a black sheep, in his words "not like the rest of my family", I've only met an uncle, aunt, his mum a few times plus been to a couple of family events in the last 4 years; we got married abroad so it was a tiny wedding of less than 10 close friends mostly. We don't live close enough to visit regularly (Manchester).
Here's the problem - his mum has had severe, sometimes-treated mental health issues for many years. She's not medicated currently AFAIK. She seems to be in spirals of getting sectioned, getting well enough to get discharged into "community housing", then getting into escalating trouble. We get calls, updates from his brother as her next of kin - sometimes she's threatened to kill her social workers, the next month it'll be that she's been arrested for being aggressive outside of a school to children, then she'll go back on her meds.. then get a police visit for sending hate mail to a neighbour who has left their bin out on the pavement for too long... some other stuff about harassing another neighbour came out (watching and making logs of when they arrive home/go out/aggressively blocking their visitors).. she's been cautioned for racial slurs against strangers and neighbours before, screaming in the middle of the street about them being terrorists, it's all so messy! She's also on some sort of watchlist at the local A&E because she turns up with minor complaints and abuses staff, one time she followed a nurse back to her car, and the police were called. Occasionally she gets given notice and has to move into alternative accommodation because neighbour complaints have been upheld - she normally lasts no longer than 18 months at one address, even when it's secure warden accommodation designed for mentally ill people. Bailiffs regularly appear to take goods that she's bought on credit and then stopped paying for. His dad is still around but they're separated, and doesn't really get involved with DH's mum aside from sometimes helping when she's been arrested/sectioned (e.g. feeding her pets, but that's about it).
My DH used to get involved in his mum's care, taking time off work to visit and help out, mediate with her carers, but it was taking a massive toll on his own mental wellbeing/finances (we can't afford for him to take emergency days off work to travel down, buy last minute train tickets etc). so about a year ago he stopped and pretty much withdrew from being involved with his mum at all.
My DH and I have had a long discussion about how to deal with our happy news and his family. We don't know if his mum will pull something odd like appearing at the hospital if she knows I'm in labour (I told him we don't have to tell his parents when it happens, we can tell them once we're out, don't worry). We don't know how to handle if she appears at our home - she's never visited.
My DH is now saying that he doesn't want our future DCs to have ANY relationship with his mum, in his words: "I don't want our children exposed to her influence, she's toxic". i'm mourning for my child having a healthy, beneficial grandparent relationship - there's going to be no one!
I've suggested we could agree to supervised, short (1hr) visits in a neutral location if we visit Brighton/his family in future, but he's saying he would prefer no contact. If his mum ever shows up where our DCs are, then to tell her directly that he doesn't think it's safe for them to be exposed to her, and ask her to leave.
AIBU to wonder what the right step here is? I'll support my DH 100% if this is his final decision but are we going to regret this? Is a broken grandparent relationship (closely supervised/monitored) better than nothing?
Sorry for this being so long 