Yes, we should try to collect the diamonds on Jupiter. This would give us a motivation to progress our space travel as well as get diamonds if we succeed. In addition, even getting to Jupiter is an astounding feat. We should definitely try, if possible, because it would be bad if we didn't.
The biggest diamonds would likely be about a centimetre in diameter -big enough to put on a ring, although of course they would be uncut. Furthermore, the recent Aug 2015 experimental observation of lightning transforming rock at the atomic level for forming a crystalline layer is consistent with this prior theory and datum of RBL that lightning can directly form diamond.
If we begin a trend of using space exploration as a means of expanding riches, that trend will never cease to exist and will more than likely become the downfall of genuine curiosity in space discovery. The only reason we should look to the stars is because it will make humanity realize how insignificant we really are in the universe. There are far too many self-righteous people bustling about with more money than they know what to do with and "Jupiter Diamonds" is the last thing we should possibly be taken aghast by. Like I said before, creating a trend of using space discovery for economical benefit WILL be the downfall of all of us. It will undoubtedly create privatization of space research and/or exploration purely for the benefit of one's bank account...How selfish is that. Space is something magnificent...Entirely misunderstood and unknown. If we can go to Jupiter, the last thing we should be worried about is the diamonds that may or may not be located there.
It is highly unlikely that Jupiter contains usable diamonds. Jupiter has an extremely dangerous atmostphere that would make it nearly impossible to reach the surface of Jupiter. As the Juno craft is demonstrating, Jupiter is not an ideal place to explore to its radioactivity and storms. Besides, even if we did manage to get diamonds from Jupiter, having to many diamonds would make them worthless. And just because we find things in space, doesn't give us the right to use them.
First of all, we don't know enough about Jupiter to determine if there are any diamonds or what kind they are. Last I heard, Jupiter was a gas giant and has no land, and I haven't heard of any diamonds on any of its moons. Also, Jupiter is very far away and we don't have the technology to take people there right now. Even when we develop the technology, it will be extremely risky and would result in lost lives.
I believe that understanding and learning about space does not give us the ultimate right to conquer it. I feel the same way about the environment--it is about respecting the world we live in. We should not be motivated by monetary value when it comes to something so vast and unpredictable. Leave Jupiter alone.