In first two generations of Blockchain, there may be no issue with security aspect. However, in third generation (e.g AION) it can occur. In first two generations, issues are related to power consumption and storage.
I suggest that you check the March 2018 issue (volume 6) of the IEEE Access (open) Journal, which has a specific topic on: " Research Challenges and Opportunities in Security and Privacy of Blockchain Technologies ". There are around 17 articles.
On top of the security issues mentioned by Horst Treiblmaier , there are issues around safeguarding private keys. Furthermore, securing the link between the blockchain infrastructure and the peripheral applications that are reading and writing from and to the blockchain is not to be ignored! In other words, protecting a blockchain-based solution is not limited to the blockchain structure only, but should also consider the whole chain of requests/responses to eliminate risks of man-in-the-middle (MITM) attacks. This is much harder than it can be thought!