I would not trade equity for design work. Unless this person will actually become a part of your team and create enough value to warrant equity. Otherwise it is short sighted to trade equity for something you need very early on just to launch. I'm sure your design/brand will evolve over time and this person will have put in a minor amount of work compared to what it will take to launch and continue a successful company. Additionally, you may be putting too much weight on your brand and not enough on what you're actually selling? However it is hard to give a more detailed answer without knowing what exactly you are doing and how detailed and ongoing the design work is. I would love to help answer your question, but need a little more detail on what exactly it is you do and what type of design/branding work you need.
Equity should be preserved for something of real value that cannot be achieved otherwise. Equity for a design partner - yes. Equity for a design deliverable - No. The benefit of equity is that the person receiving it becomes invested in the success of the business and will continue to work to help it grow. A graphic design is static - once its delivered the value exchange is complete.
I won't trade equity for graphic design.
Design is important. In fact, a good design can make a world of difference, however, unless you are literally selling design, I won't part with equity for it.
End of the day, having a good design is like having vitamins. It keeps you healthy, makes you feel good - and that might be a differentiator for some.
But, I would trade equity for a painkiller. Something that solves a real pain point. It could be engineering, sales etc.
I would say no. Giving equity means you give a part of your company to someone. Hence, you give a little bit of control of your company to someone.
If it were me, I'd give equity only to people whose engagement is vital to the future of my company. Namely, people I want to see in my company for the long term (i.e. use equity as a retention measure) and whose contributions are key to the well-being and functioning of my company (aka, not for people who can be easily replaced or whose job can be easily given to someone else).
If cash for payment is a problem, I suggest you find other ways to finance your design work. Call me if you need some ideas for that.
I wish you much success and fun developing your brand!