Those who want to invest, of course, will not want it to remain constant. Even if it is falling, the fact that it is moving keeps the market energetic and there is a constant flow of money in the market. As the value of Bitcoin increases, the number of people who want to invest will also increase. Therefore, more money will be involved in the market. Obviously, the constant value of bitcoin will not make anyone happy.
I don't actually think that Bitcoin's price will remain constant even if we have reached or mined all of it but rather we can see a less volatile market on bitcoin the closer we reached the maximum amount. If you think about it, it will not be bad on the market as there will be a constant flow as there is still volatility on the price and it will be good for it's main purpose as a mean of medium for peer to peer transaction.
As of now, most of us are more concern on how much volatility the price of bitcoin is as we often look at bitcoin as an investment opportunity.