I know they were essentially personal servants to officers, but how far did that go? Did the participate in offensives?
It would depend on what rank the officer was, and what his position was. If it was a "non-combat" role like a medical officer or an engineer, then their Batman probably would only see combat going to the lines to collect wounded, fix roads and lay duckboards, etc , but such work could be just as dangerous, with snipers, artillery and pockets of enemy resistance always being an issue.
If the officer was any rank between Lieutenant and Brigadier General, and was in a combat role (i.e. infantry, cavalry) then their Batman could expect to be at the frontline, and likely in combat, quite frequently. 78 British generals were killed by enemy action in the Great War, and losses among junior AND senior officers were pretty high.
Notably Colonels, who would be in charge of Battalions, were in the line of fire quite often. Men like 'Frank' Maxwell VC, who won the Victoria Cross in the Boers War, and led the successful capture of Schwaben Redoubt in Sept. 1916; Adrian Carton De Wiart, an absolutely remarkable man, who also seemed to scare DEATH, because he was wounded and in dangerous situations countless times, but NEVER DIED (he also got a VC); and Lt. Colonel George Augustus King, DSO & Bar, who commanded the New Zealand Pioneer Battalion, and was awarded the Croix de Guerre by the Belgians for his service in the 3rd Battle of Ypres ('Passchendaele'), and was killed during that battle. In fact, he was so popular among the New Zealanders, that some Māori's from his Battalion performed a waiata tangi dance rite at his funeral (only reserved for high ranking Māori chiefs!)
Considering the amount of danger that British officers were exposed to in their line of work, it's safe to say that the life of a Batman was just as hazardous!
Tommy, by Richard Holmes Mud, Blood and Poppycock, by Gordon Corrigan Bloody Red Tabs, General Officer Casualties of the Great War, 1914-1918, by Frank Davies & Graham Maddocks Three Armies on the Somme, the First Battle of the Twentieth Century, by William Philpott
What do you mean by Batman? I've never heard that term, and I assume we're not talking about the caped crusader.