Which cryptocurrencies are considered base coins? Are these the only base coins: BTC, ETH, BTH and LTC?

Base coins are those that you can buy directly (USD -> base coin). Alt coins such as NEO, XRP and EOS require first going through a base coin.
In the context of defining basecoins as those that change directly with USD, there are actually quite a few. The 4 you've listed are available through Coinbase, which is currently the most popular fiat to cryptocurrency gateway.

However, there are many exchanges that still accept USD/EUR/JPY/KRW deposits with some level of verification. For example Bitfinex allows for fiat deposits after verifying your account, and they have 20 or so crypto assets (including EOS, XRP, NEO that you mentioned). I'm sure there are many more exchanges that allow fiat deposits, even if crypto deposits are growing increasingly popular for legal reasons.