In cryptography, randomness is mostly about being unknown to any attacker. Any attacker could observe your download of these random numbers, and now they are not really random since the attacker knows them too.
There is an article from Crypto'97 which describes something which looks like that: it is a secure key exchange system which defeats memory-bounded adversaries. This assume a shared source of randomness which broadcasts random numbers; anybody can listen to it, including any attacker. Two parties who wish to get a shared key record just a few chunks out of the random bits, noting the position of each chunk in the stream (say, the exact emission time). At the end of the day, they send to each other the list of positions; if they recorded enough, there is high probability that there are a few chunks they they both recorded, and they can use these as a shared key. On the other hand, an attacker willing to learn that key would have to record quite a lot of the stream in order to have a fair chance of having all the chunks that the two parties have in common.
This key exchange mechanism is safe as long as the random source broadcasts data at a very high rate, so that recording all of a day's broadcast is not doable. So it should be in gigabytes per second or so.
In a way, using the numbers from
random.org is like using this model: you record insecure random numbers, and at the end of the day you assume that you could get some that the attacker failed to record himself. To get a key out of your pool, you could for instance hash the whole of it with a secure hash function such as SHA-256 (don't pick numbers out of the pool; instead, use a hash function, which will not "forget" any entropy). Still, in practice, this fails on several accounts:
The Web in general, and your Internet access in particular, is nowhere near fast enough for this. An attacker can easily record terabytes of data, so you should download much more than that. Your ISP will not be pleased (unless your contract includes a quota, and gigabytes beyond the quota are billed, in which case the ISP will be thrilled).
random.org itself could be an attacker, and feed you what are claimed to be "true random numbers", but are actually the output of a pseudo-random number generator. You would not be able to see the difference. But this would allow the attacker to rebuild the random numbers you got (all of them) at will.
An active attacker could alter the numbers you get, in effect reducing the problem to the previous point.
So, to make the story short: no,
random.org is not useful to security or cryptography.