Are there any techniques available today that would make it considerably easier to anyone attempting to translate it today (as if they were trying todo it for the first time)? Otherwise would the process be the same as it was for the first person who managed to translate it? Would computers be of use and if yes how?
I think there are two big innovations that we would look to today.
Computers able to analyze millions of patterns in minutes
Cryptographic techniques that use what is known about patterns of a language to discover regularities in the seemingly random noise of a cipher
However, the fundamentals would be the same. You always have to start from a fundamental understanding of language. You always need a starting point. And you also need a decent corpus to use whatever quantitative techniques you have on. That's why so many scripts with only fragments and no language available remain a mystery.
But even parallel texts may not be enough. The difference between cryptography and Rosetta stone is that in the first case you're dealing with at least one known language. (That's why Germans couldn't crack Navajo communications.) And translations are never exact parallels. So one fact Champollion relied on was that 'cartouche' - oval around some text - was used to indicate royal names. That gave him an anchor from which to work backwards. But he still had to come to an understanding that the hieroglyphs are not strict ideographs and then rely on his in depth knowledge of Coptic (he kept a diary in it) to make something of it.
None of this could be achieved by pure pattern analysis but it could certainly speed up some aspects.
Similar thing happened in the decipherment of Hittite which relied on sound philological work once a large corpus of the language was found in a known script.
However, no computer method can help with the decipherment of scripts like Linear A because of the tiny corpus from which to work.
Even where a much larger corpus is available as with the Indus valley Harappan script, it's hard to know what to apply modern techniques on since it's not even clear the script represents a language in the conventional way.
I think there are so many avenues to explore in this area that its hard to answer this question. I believe there are many ways computers could be used in the decipherment.
Kevin Knight's research is particularly relevant in my opinion.
You may be interested in these papers, that are in that direction.