I suspect you may be clicking on the download button when the register is open.
By default this starts an internal http implementation (a limited browser if you like) This is very much in the early stages of development and is known to not work with many institutions at this time. If you are using this button hold down the control key (command key on Mac) when you click on this button and the system browser will start instead.
It is my understanding that Chase supports Direct Connect for automated downloads, and if you set this up the download button will start this process in preference.
Take note that you must first authorize your accounts for desktop software on the Chase website.
The option to do so is hidden fairly well on the Chase website.
Click the head-and-torso icon in the upper right-hand corner.
Click 'Profile & Settings'.
From the resulting menu on the left, select 'Account Safe'.
You should then click on 'Desktop Software'.
I'm not 100% sure of what the next step is as my accounts are already authorized but I'm sure it's self-explanatory.
Once you've authorized your account, run the Set Up Online Banking routine in Moneydance and try to connect. Take your time going through the various steps of the routine - there is no room for error.
Afterwards try to connect to Chase by either clicking the menu item, using the appropriate key-combo or by clicking the 'Download' button in the register.
If you receive an error message pertaining to verifying your identity, look to your inbox for instructions on how to do so. Don't worry, it's a simple matter of logging on to Chase and clicking a link.