Hi Jack
Three suggestions in Alicante city:
1. Benelua, a suburb just to the west of Alicante city centre, might tick some of your boxes.
It isn't very expensive and it is a good base for travelling around if you don't have a car. The train station is 5 - 10 minutes walk away (depending where you are in Benelua) and has trains to places like Madrid, Barcelona, Valencia, Murcia and Elche. The bus to the airport stops outside the station (20 minutes there, 40 minutes back, on a circular route). Another 5 minutes from the station there's the start of the electric tram line, which runs along the coast all the way up the Costa Blanca. There are plenty of local buses and the main bus station is 10 - 15 minutes walk from Benelua (e.g. for the bus to Gran Alacant and Santa Pola).
It is reasonably quiet apart from fiesta time (where everywhere in Alicante seems to have noisy celebrations) and seems pretty safe. There aren't any great views. It is just an ordinary Spanish suburb but that does mean the shops and restaurants and bars are open all year, not just in the tourist season. In fact it probably has more shops than just about any other area in Alicante, so living day to day is very easy. Nearly everyone there is Spanish and, like a lot of people in Alicante, most don't speak much English, so it would help if you speak at least a bit of Spanish. Having said this people seem very friendly and helpful, and sometimes, if they can't speak English they'll find a colleague or relative who can translate - or there's always sign language, Spanish classes or Google Translate.
It is about 20 minutes walk to the start of the marina and about 30 minutes walk to the beach - or you can catch a bus.
2. If you want to be nearer the sea, without having to pay a lot of money for a central location, you could try the area just to the east of the Santa Barbara castle, like Calle Virgen del Socorro, which is nearer to the beach and the marina and the old town than Benelua while also, (in the other direction) in walking distance of an electric tram station and the big Plaza Mar shopping centre.
3. One time we rented an apartment in Calle Portugal. I'm not sure what the area is called but it is half way between Benelua and the city centre, so medium price and quite a convenient location wherever you want to go, in the city or further afield.
Good luck with your search. Other people may be able to suggest different areas in the city.
Mike