Crap, hurry with the segwit devs!
The only way an increase in the blocksize would relieve this situation is if the spam transactions made it into blocks.
That would allow attackers to cause a rapid growth in the size of the blockchain resulting in more resources needed to run a full-node without legitimate transactions increasing to justify the increased resources.
Even if we had 16 MB blocks my hope is the miners would not be including these junk transactions in blocks.
Including them in blocks serves no useful purpose to the bitcoin community.
All transaction must included in blocks, even it's only spam transaction.
What miners should do is prioritize big transaction or any transaction which not look like spam transaction.
So, regular user won't be affected too much with transaction spam.