~でも in だれでも should be equated to であっても "even to be", so だれでも = だれであっても = "no matter who (s/he) is". Translating it as "anyone" is indeed a clever way, as it puts stress on that no one is exception. Grammatically, the particle is under the same category with は and such (取り立て助詞 "focus particle"), so here you can take it as a sentence adverb.
だれも{HLL} in だれもが is a special pseudo-noun, which obviously derives from だれも{LHH}. Since the particled phrase だれも, literally "whoever", is only paired with negation in the modern language, we have had to work out a synonym usable in a positive sentence, so this weird form has born. Consider it as a special form of "whoever" in the nominative case.
Now, the intricate portion is the predicate 楽しむもの(じゃないか). It is not accurate enough to interpret using the English present tense like "something (one) enjoys". Japanese present form is by default future-oriented, thus the strictest reading is "something to enjoy".
With that, if you put だれでも楽しむもの, it will literally yield "something to enjoy, no matter who (s/he) is". This would be used to tell "everyone should or is likely to enjoy rather than not" or "everyone should enjoy rather than doing other things", which is somewhat off the point in this argument. It does not mean the combination is unnatural, for example you can say:
漢字はだれでも習うものじゃないか
On the other hand, だれもが楽しむもの(だ) is "something for whoever to enjoy" (= something to V + whoever enjoys), which is more neutral in nuance.
However, I don't think だれもが楽しむ is the most natural expression in this case, because it suggests everyone now does or is going to enjoy (which is a bit too strong remark unless in a manga circle). In the context (see OP comment), what the speaker intends is probably that it is open to everyone who wants to enjoy, that would be:
だれもが楽しむようなもの
(だれでも/だれもが)楽しめるもの
(だれでも/だれもが)楽しめるようなもの