One day it may be possible to walk/cycle through Leith Docks to Marine Esplanade. There may even be a decent path behind the sewage works.
Until then the choice is Seafield Road or the path that starts on Leith Link. The second part of this can be on the shared use pavement section of Seafield Road.
Upgrade in progress –
At present it’s possible to explore the area east of Leith Docks and ‘enjoy’ the sights (and smells of the sewage works).
Once the Docks were pretty much ‘open’ for access and exploring. Fear of rabies stopped that. This is the eastern entrance –
The other entrance/exit is at Bath Road, which is actually very close.
If you ask Google to plot a route from the Esplanade to B Rd. this is what you get –
Curiously it states “This route has restricted usage or private roads.” But the plotted route isn’t the one that is actually open to authorised users!
This is the current reality of Albert Road –
Similar at the western end. It could be open for walking and cycling.
Creating a route between Ocean Way (where Port of Leith Housing Association is due to build houses) to Bath Road would ‘complete’ access through the Docks avoiding the working port.
Image from Edinburgh Local Development Plan
Next: Seafield Prom
EdProm: Edinburgh Waterfront Promenade project 
Planning an ‘Evening at the Proms’ for the Edinburgh Festival of Cycling. Monday 16th of June proposed. This might not do the Seafield route but will go in along Shore and onto the Leith Links to Seafield path which should be finished by then. Purpose is to look and admire (hopefuly) the good bits and endure the poorer bits. Potential is there and with more people due to live in Leith they need some alternatives to driving everywhere!