"Do Klingon ships have escape pods?"
"Yes, it's where we keep the cleaning supplies."
- Old joke, my wording, original source unknown to me
Any answer is going to be heavily era, source material and ship class dependent.
TV & MOVIES
Enterprise tv show - Klingon escape pods exist, they are used
You have noted in your question that in Enterprise there appears to be conflict. In ENT 2.25 and 4.06, Enterprise rescues people in Klingon pods, one Klingon, one human.
In 1.14, it is noted that at least some small Klingon ships don't carry escape pods.
Star Trek (original) tv and films, Star Trek Animated - no information either way
In film and tv media through the '60s through '80s, I don't recall any mention of a Klingon escape pod and haven't found any references they exist.
Deep Space Nine - Klingon escape pods exist, a Klingon once used one for its intended purpose, their acoustics are very good for Klingon opera
There was one in 1999/2000 in DSN 7.17 as noted in another answer. The Klingon was Worf, and I think he was the one who said they had good acoustics. This doesn't definitively tell us that Klingons used escape pods--for anything other than singing--but given the often utilitarian nature of Klingon equipment, it doesn't seem likely that Klingon ships would have them if their only purpose was "plot device" or "let Star Fleet officers survive" especially as ship designs often predated alliance with the Federation.
Voyager, Next Generation - no information
I haven't found any reference to Klingon escape pods.
FASA RPG - inconclusive that they exist and are used but plausible
No escape pods are shown on the D7 blueprints released by role playing game publisher FASA.
- given the size of a D7, lack of space is unlikely to be an in-universe reason for omission which would suggest escape pods don't exist and aren't used.
- Unfortunately, this is not definitive because:
- the FASA Enterprise deck plans also omit escape pods.
- A large Federation ship like Enterprise lacking escape pods is seriously at odds with our expectations for a Federation vessel, so much so that we have to regard the omission as an error.
The Klingon Ship Recognition manual notes that the D7 command pod (the bit at the front of the boom including the main bridge) existed as a form of life boat that could be jetisonned and which could keep the crew in it alive for a year. (The deck plans shows engines at the rear of the boom, not the pod, which is at odds with its described ability to propel itself.)
- This is repeated for several Klingon ships.
- Escape pods are not mentioned.
- But that they have a mechanism to detach a life boat section says that escape pods aren't inconceivable.
- Klingons value personal survival.
Allied with the notion of conflict as a positive experience is the basic need for survival. ... Common beliefs [held by Federation citizens] notwithstanding, the typical Klingon does not fight and die simply for the sheer sport of it.
"Klingon Psychology", Klingon Intelligence Manual, FASA
Especially following the emergence of The Next Generation, FASA's Klingon innovations, largely inspired by the novels of John Ford, and the rest of the FASA work was ignored or altered and--if you happen to believe that Paramount or CBS or whoever can declare things to be canon--declared non-canon.
Star Trek John M Ford Klingon novels - Klingon escape pods might exist, if they do they are used though sometimes their use may arouse suspicion
“We have one who is not forgotten,” Krenn said. “His name was Kahless. When his ship was dying, he had his hand bound to his Chair, that no one could say he left it, or that another had been in the Chair at the ship’s death. Then all his crew could escape without suspicion, because Kahless had taken on all the ship’s destiny.
The Final Reflection, John M Ford
The above states that Klingons escaping a crippled ship was completely acceptable to them if someone stayed behind. That it was "without suspicion" indicates that other times crew might escape with suspicion--but that's still escaping.
A careful skim through the book tells me that that passage is the whole of all mentions of Kahless in the novel besides using his name as an oath "Kahlesste kaase" (Kahless's hand).
Other sources (notably FASA) say Kahless died in battle but there's no indication in the novel of where the battle took place but it was before transporters were used.
- Kahless is considered to have died long before the novel is set (at least two centuries before the Five Year Mission).
- The novel relates events just before Kirk is born.
- In the novel, transporters are so new that the Klingons don't know:
- if the Federation have yet invented them (they haven't), or
- if the Federation has learned that the Klingons have.
As it was a "dying ship", and assuming it was during a battle, it doesn't seem likely that:
- the ship landed on a planet so the crew could disembark or
- there was time for other ships to dock (not that it seems likely Kahless died while he had allied ships around anyway, it seems like a one-Klingon heroic last stand).
There aren't enough shuttles on ships to carry the entire crew complement,
- but there may have been heavy casualties so only a few shuttles were needed.
Shuttles or escape pods are the most likely means of escape. Escape may arouse suspicion (cowardice? waste of resources?) normally but this tells us that escape was an option, even if one generally frowned upon.
ADB Star Fleet Battles - Klingon escape pods exist, they are used
In Star Fleet Battles, there are four Klingon Fast Attack Ships. It is standard for fast attack ships to have escape pods.
Last Unicorn Games RPG - Klingon escape pods exist, they are used
Yes, Klingon ships of the TNG and TOS era were listed as having four-Klingon escape pods, enough for all the crew I think.
I reget not having had the Klingon Bird of Prey Owners' Workshop Manual at hand when I started this yesterday but since I have extensively added to and reformatted I'd ask you to please see the accepted answer for details, the short version of which is: "exist and used".
Fandom - Don't exist, or, Exist but not used for escaping
According to some fans and a trope website (looked for it to reference but couldn't find it again), many fans say they don't exist because they're non-Klingon, Klingons would rather perish in battle. My comic answer at the start which says they aren't used as escape pods is a joke that arose sometimes in the 90s at latest, again the idea being that escape pods are un-Klingon.