The answer depends on where you are, what your local energy prices look like, whether you already have gas coming into the house, whether you're going to be there long enough for investment in a new system to justify its cost...
In my area, gas and oil have historically alternated as most cost-effective, with electric heat always being most expensive.
When I bought it, my house had a 1945 oil boiler, inefficient by modern standards and with a slightly leaky tank as well. I budgeted for replacing it when negotiating the purchase. I personally prefer gas and already had gas coming into the house, so that's what I chose -- a direct-vented high-efficiency gas boiler, with new pumps, feeding the existing radiators.
(Actually, I was very interested in the small gas-fired cogeneration plants starting to be sold as heating systems ... use the gas to run an engine which runs a generator, sell the electricity to the grid and use the "waste" heat to heat the house -- but for good or ill, they were only available for forced-hot-air systems and I wasn't ready to replace everything or deal with having only one or two trained technicians in the entire state to maintain the thing. So I'm just selling solar to the grid.)
One thing to keep in mind when upgrading a heating system: Old units drew combustion air from their surroundings, drawing (cold, in winter) replacement air into the house. New units are direct vented, meaning they draw combustion air directly from outside, so you don't lose as much heat out the exhaust. But their greater efficiency also means less heat lost to their surroundings, so the basement may actually become cooler rather than warmer unless you remove some pipe wrap to get more heat into that space... or add a zone and radiator(s) for the basement.
Dangerous? If it malfunctions and an over-pressure event occurs, that's dangerous. Whether or not a functioning gauge will provide early warning or not would depend on the experience/knowledge level of the gauge reader and the nature of the malfunction.
Are you required to have a functioning pressure gauge? Most likely you are but it may depend on the duty rating (BTU), type of occupancy, and where you live. Some boilers in some states require regular documented inspections which include a pressure check. Inquire with your local code enforcement agency, although you may not like what you find out about responsibilities attached to steam boiler operators.
But the rules are written in somebody's blood, just do an internet search for boiler explosions and you will see.
Best Answer
Eventually if water continues to fill the system, it will fill the pipes, and eventually come out the vents on the radiators. I saw a house that got flooded when someone was adding water to the boiler manually, forgot to turn it off, and left - made a big mess. Obviously that would take a long time with a slow leak.
I am not sure whether this also creates a safety issue.
There may be an automatic fill valve that isn't sealing off properly when the water reaches the full level that can be serviced or replaced. I believe these can also be retrofit if there isn't one on your furnace.