How cryptographically secure was the original WW2 Enigma machine, from a modern viewpoint? If cryptanalysts today were to crack the original Enigma machine, “how fast” or “how easily” could they do it? What methods would they use?

The original cracking was significantly helped by operator mistakes, like always including the same header, always including a weather report, and finally, a long message of LLLLLLLL… being sent. Let's assume no such operator mistakes happened.
What methods would they use?


Since WW2, we know the security of Enigma machines was weakened by the reflector, resulting in two problems:


No difference between en- and decryption, which means that if K ↦ T, then T ↦ K.
No letter can be encrypted by itself because electricity can not travel the same way back, which results in a reduction of encryption alphabet.


Also, history has shown that Enigma had another security problem: the contradiction to Kerckhoffs' principle "The security of a system should not depend on the privacy of the algorithm. It should only be based on the secrecy of the key.".

Enigma got it wrong because:


Security of Enigma depended on wiring of rotors,
Wiring was part of algorithm and not part of key, and
Wiring never changed from 1920s until 1945.


(source: https://www.infsec.cs.uni-saarland.de/teaching/SS07/Proseminar/slides/maurer-Enigma.pdf)

Since you're asking about today, I will therefore assume the fact that we know about the internal workings of Enigma since WW2.

Based on your assumptions and the known Enigma weaknesses, one of the well-known attacks to break Enigma is using the Index of Coincidence (short: IC), a statistical measure of text which distinguishes text encrypted with a substitution cipher from plain text.

Let's put Enigma into the formula: $IC = \frac{{\textstyle \sum_{i=0}^{25} } f_i(f_i-1)}{n(n-1)}$ where $x_1,...,x_n$ is the string of letters and $f_0,...,f_{25}$ the frequency of the letters in that string. For a random string that boils down to $IC ≈ 0.038$ and for a natural language $IC ≈ 0.065$.

I'll skip the details and simply give you some raw numbers while summing up how IC can be used for an attack on Enigma:


Find the rotor order
Trying all rotor orders and positions searching highest IC takes $60*26^3 ≈ 2^{20}$ operations.
Approximation to rotor start positions
Starting with a rotor order from step $1$ and $w$, trying all rotor positions and ring positions for 1st ring only, again searching highest IC. That takes $26^4 ≈ 2^{19}$ operations.
Find ring and rotor start positions
We have the 1st ring and 1st rotor start positions, approximations
for other rings and starting positions from 1 and 2. First search positions for the 2nd ring and rotor, then use the same procedure for last remaining rotor. This will take $26^2 ≈ 2^9$ operations.
Find the plug settings
We have the rotor order, position and ring positions. Now we can use IC as statistical test again, deriving of the trigram information of the underlying language.



  how fast/ how easily could they do it? 


Generally, that's hard to say without having any specific information about available resources you are assuming an attacker would have. 

A single computer might take a little while depending on how old it is (in terms of CPU power), but things like distributed networks will probably make you cry by successfully attacking Enigma in a few blinks of an eye. If you happen to have any specific computing setups in mind, you can take the provided number of operations mentioned above and simply do the related maths to calculate the related speeds.

And last but not least - to answer the title of your question too...


  How cryptographically secure was the original WW2 Enigma machine, from a modern viewpoint?


Totally insecure. The weaknesses described above provide ample proof to that claim.

On the other hand, we can't really compare today with the situation during WW2. They used machines that were mainly mechanical (eg: Bletchley Park Bombe), while we have computers with powerful CPUs. WW2 cryptanalysts  would probably have killed to get their hands on such a futuristic "wunder Maschine" which we use to do simple things like surf the internet.

Nota Bene: If you're interested in the maths behind Enigma, you might want to check on "The Cryptographic Mathematics of Enigma" by Dr. A. Ray Miller, The Center for Cryptologic History – National Security Agency.

The above publication certainly provides more insights related to the internals and maths behind Enigma than the usual NSA "freebee" stuff (PDF) which tends to restrict itself to talking about the history instead of cryptography.



Edit: The last two links are now offline and seem to have been removed from the nsa.gov website. As they are still available via web.archive.org, the links have been updated point there.