The "penalty" is two fold it seems, either effecting your Ship or you Bridge Officers (or both) and range from Minor to Major to Critical.
There are two locations where you can go to have them "repaired," DS9 or SB1; and depending on whether it's your ship or your BOs you will need to speak with an Engineering Officer or a Medical Officer. At SB1 the Engineering Officer (Repair Ship) is located in the Shipyard close to the outside window and the Medical Officer (Treat Officers) is in the Personnel Section near the outer wall away from where you get more BOs or train new skills. At DS9 the Engineer is in the Shipyard and the Medic is in the area where the Exchange and Bank is located.
Additionally you can buy "Components" (to repair your ship) and "Regenerators" (to treat your BOs) to keep in inventory so you're not flying all over the galaxy every time you die and have to fix/heal everything. On SB1, the Components can be purchased in the Shipyard on the lower level from the NPC "Requisition Ship Components" towards the center of the station. Minor Components cost 1,000 EC and Major Components cost 8,000 EC. You can buy Regenerators from the NPC "Requisition Devices" located near the Tailor; Minor Regenerators cost 200 EC and Major Regenerators cost 1,600 EC. I was not able to find either the Critical Components nor Regenerators at SB1.
On DS9, I found both Minor and Major Regenerators from the NPCs "Device Merchant Dolgan" and "Device Merchant Tohmoe" (next to each other) on the Promenade for the same prices as SB1, however I could not locate anyone selling the Components for ship repair (I must have over looked them).
Again, this is the preliminary information that I have gleamed from being on Tribble. I have not yet experienced any Death Penalty and will be updating this as I learn more. Please add any information you have or make any corrections that need to be made.