There are a few things you can do to help build trust to get new deposits:
1. Inbound marketing, you need to create a strategy for the long haul to build value and set yourself up to be more than just a bank. This deserves it's own question and answer.
2. Make sure to promote your deposts are FDIC/NCUA insured.
3. Many people were leaving big banks for some of the business practices they engage in, trying to focus and highlight that you are different, that you don't do what other banks do and take care of your customers will help you attract those looking to switch.
4. In addition to having basic banking features, you need to deliver additional convenience features that most banks offer now - online banking, mobile deposits, fee-free ATMs, etc.
5. You really need to figure out your ideal target market, find something niche you can focus on and attract them. Without a huge marketing budget, trying to market yourself to general public will not yield results. If you focus on a niche target market, identify their pain points with banks, and then try to offer an alternative that fixes those pain points will get you those customers. Once you get more customers, word of mouth will spread and you will be able to compete to a wider audience.
6. I understand the reasoning behind getting deposit customers, to fund the loan business, but that's not going to help you get AND keep customers. You do need to be a good bank, with good customer service, with a good portfolio of features.
If you get one thing out of what I wrote, that should be this: Figure out what your ideal target market is, identify their pain points with banking needs, and position your bank to fix those pain points. You might want to start with your local neighborhood, and work from there.