It's hard to get absolute numbers, but comparing relative numbers of speakers is possible. Numbers used in other answers are not comparable, because they come from different sources. So I decided to use one source: LinkedIn profiles. They have a fixed format for specyfing language proficiency with 5 levels: "elementary proficiency", "limited working proficiency", "professional working proficiency", "full professional proficiency" and "native or bilingual proficiency".

I took a list of some well-known conlangs and ran a lot of Google queries in the format: language name + proficiency level, e.g. "esperanto * elementary proficiency" , "klingon * limited working proficiency" , "volapük * professional working proficiency" , etc. I restricted each search to the LinkedIn domain with site:linkedin.com and counted the results - each result meant one person who listed a particular language at a particular proficiency level. I had to go to the last page of the results, because the numbers reported by Google on the first page are widely inaccurate.

Here's what I got (note that Elvish is very often listed as a seprate language, even though it basically means either Quenya or Sindarin):

native full prof. prof. limited elementary Esperanto 59 118 148 252 483 Klingon 75 44 50 83 205 toki pona 3 6 10 11 29 Lojban 1 1 4 16 45 Elvish 12 5 5 16 26 Quenya 2 0 1 17 25 Sindarin 5 0 3 12 19 Na'vi 2 1 1 7 8 Volapük 2 1 1 4 6 Interlingua 0 3 1 4 1 Elefen 0 0 2 2 0 Ithkuil 0 0 0 2 2 Ido 0 0 1 1 0 Solresol 0 0 0 0 1

That's the raw data. Many different interpretations can be made, here's mine.

Probably the biggest surprise is the very high position of Klingon. Especially the "native" column makes me think that many Klingon entries are just jokes. There's simply no way there could be more native Klingon speakers than native Esperanto speakers. I would say that the whole group of artlangs (Klingon, Elvish, Quenya, Sindarin, Na'vi) is quite specific. These artlangs were never meant for actual communication between humans. Moreover, many people may learn them just because they are fans of Star Trek/Tolkien/etc and not because they are interested in langauges. Therefore, many speakers of artlangs may have a lower standard for what "proficiency" means.

Despite all this, we can't ignore the high position of Klingon. Wikipedia says it had "around a dozen fluent speakers" in 1996, and Yens Wahlgren's Bachelor's thesis says 20-30. These numbers seem underestimated. Klingon Language Wiki lists 38 "Klingonists of note", most of whom must have an advanced level in Klingon. It is made clear that it only lists people who have contributed in a special way to the community, so we can expect that the list of the speakers is longer, even in the Klingon Language Institute. Moreover, Star Trek is widely known and has fans around the globe - any dedicated fan with language learning aptitude may learn Klingon on his/her own without any association with KLI. Apart from that, among 604 respondents to the Klingon Questionnaire there were 93 who assessed their level as intermediate, 49 upper-intermediate and 16 advanced.

For comparing the number of speakers of the languages, I think it makes sense to sum for all the levels, except for "elementary proficiency". People may say they know a language at the elementary level if they just read a bit about it, and can't really use it. Therefore, I summed the number of people listing each language at the "elementary working proficiency" or higher. Here are the results:

Esperanto 577 Klingon 252 Elvish 38 toki pona 30 Lojban 22 Quenya 20 Sindarin 20 Na'vi 11 Volapük 8 Interlingua 8 Elefen 4 Ithkuil 2 Ido 2 Solresol 0

Esperanto has clearly the highest number of speakers, by any count. Some languages can grouped together, as the differences between them are not significant. As far as non-artlangs are concerned, toki pona and Lojban are clearly on the second place, with more speakers than the rest. The ordering is as follows:

Esperanto toki pona, Lojban Volapük, Interlingua, Elefen, Ithkuil, Ido, Solresol

If we compare Esperanto with artlangs, the ordering is:

Esperanto Klingon Elvish languages (Quenya & Sindarin), Na'vi

For the reasons I explained above, it's very hard to compare the number of speakers of artlangs to speakers of non-artlangs. My very rough guess is that the number of Klingon speakers has the same order of magnitude as the number of toki pona or Lojban speakers (that is, one or a few hundred). Elvish languages have fewer speakers, so the order of magnitude is probably similar as with the rest of the languages. It means that the order should be roughly like this: