No. The security of a wallet is NOT depending on the transaction fees.. It is depending on how the wallet stores/manages the private keys.


There are multiple types of wallets (from most secured to least secured):

• Cold wallet storage / Paper wallet:
  Private keys stored on an offline computer NEVER connected to the internet and only used for sensitive things (transaction signing)
  Very inconvenient, but also very secure
  
  
• Hardware wallet:
  Physical device. Private keys stored in an isolated offline environment. Can be plugged in into computers and used safely without the fear of private keys getting leaked.
  Easy to use and very secure. But costs 70$+
  
  
• Desktop wallet:
  Software wallet on your desktop. Should be open-source and private keys / wallet file should be encrypted.
  Easy to use and only as secure as your computer is
  
  
• Mobile wallet:
  The mobile equivalent to a desktop wallet. Can be a bit more or less secure. This depends on multiple factors
  
  
• Web wallet:
  Definitely the least secured wallet. Your private keys are stored on a server. Not recommended.
  
  
• Exchange / 3rd party site:
  Can not be considered a 'wallet'. 3rd party has full access to your money. Should be avoided.