Chances are high. Just don't give them any automatic weapons or rocket launchers or grenades.
Still the best (and IMO the only) use for them is as walking storage.
In fights they often stand in your line of fire and you can't use the big guns.
In small rooms they more than once they stand in your way.
Be prepared that you need much more Stimpacks and ammo.

Using sfall you can edit INI file to take control over your companions - they will not do a step without your permission... :>

Unless someone modded it in, companions in Fallout 1 can't wear armour, so you'll have to micromanage them a lot to survive hordes of mutants.

I'm surprised you went through by yourself, as there's no limit to the number of companions you can have at any one time in Fallout 1.

Although having said that, there aren't that many companions to choose from either. I won't mention them here in case you want to find them yourself, but from memory there are four in total.

The controls are limited, and they wear only the armor they start with. You can improve their weapons, but there are certain guns they can't use based on the animations of the characters (so it may not be totally obvious which ones they can and cannot use, as some pistols use the SMG animation for example). In general I think most are limited to pistols, rifles and SMGs anyway - not sure if they're even capable of using heavier weapons (not that you'd want them to).

In F2 you had more control, but in F1 you can only specify their distance from you IIRC - and I think that's mainly the distance they keep from you when not in combat.

As mentioned above though, you can get sfall for F1, which I think has an armor mod included (or you can find one separately) - so armors can be upgraded. It also allows direct control, but personally I think that's going too far - I like the fact that they're "under their own control" so to speak. Not sure if sfall for F1 adds Fallout2-style controls for the team - it might be full control or default F1. TPR might be able to answer that...

I've played through Fallout 1 several times and only once did I manage to get ONE npc alive through the military base... Somehow. They were all dead by the final boss fight, unsurprisingly, simply because you need Power Armor for like the last 10% of the game and they can't be equipped with any armor you have on hand. In fairness you really don't need any support by that part of the game (atleast you shouldn't if you found all the upgrades). I'm going through FO2 right now and I can definitely say that they have a much easier time surviving if you give them proper gear and weapons and I can see them surviving through the end of the game, but good luck achieving that in FO1 for reasons mentioned already.

In FO1 I managed to get all except Dogmeat to survive the military base (he couldn't get enough of those forcefields! You'd think he'd learn that they're dangerous after hitting them the first few times...). Getting the others to survive did involve me putting my own character in the line of fire and keeping them way back. Also I went in with the robe first, then only went hostile once I'd visited everywhere (but before starting the self-destruct). That way I could ensure that my companions were out of the line of fire for most of it. Also I don't think I fought the final boss (cathedral), again using the robe and then activating the bomb instead.

Now I can't remember if I left them outside these places prior to going in, or whether I was able to take them with me. I know that similar situations in FO2 people attack you if you have followers (i.e. your disguise won't work), but I think in FO1 this wasn't the case. Of course you can't tell Dogmeat to ever leave your party, so he's probably going to die.

There are actually people on the web who've dedicated (far too much) time to keeping Dogmeat alive - trapping him in a room in the military base (since he can't open doors) seems to be the best way. I couldn't be bothered going that far though.